The following is a presentation of TFNN. Time to talk about your health. Living a primal lifestyle. You know, we have Tom from Tampa on the phone. Hey, Tom, good morning. It's bright and early now, huh? Hey, hey, thanks for coming, Tom. How are you guys doing? Good. Hey, your new player's outstanding. I'm telling you, Matt is outstanding. It shows the primal edge. I love that stuff. I mean, I've been on it now three, four months, man. I mean, it's just, I can't get over how good I feel. Primal edge is, you know, people are raving about it. People who are trying it, they know because you can feel it. Would not be without it. Call now toll free at 1-877-927-6648 internationally at<PHONE_NUMBER>. Now your hosts, Niko Dahan and Paige Clark. Well, good morning. Welcome to Living a Primal Lifestyle. Well, we explore a return to a more balanced, natural, wild world. I'm Paige Clark, by the way. Yeah, and we do this to recover our natural health, regain our rights and freedoms, and I'm Niko Dahan. And it is a rainy day and Monday type day. We have had a lot of rain, 76 degrees in downtown St. Petersburg, Florida. Yeah, it was a wet coming down here. We needed it. It's greening everything up, though. It keeps the temperature low, and hence we're going to have like 85 to 87 all week. Nice. Because of the rain. Nice, nice. That's kind of good. And then we have a new newsletter. In this newsletter, you get it into your mailbox how often? The 15th and the first. The 15th and the first. And it's got clickable links to all the stuff we're talking about that'll keep you healthy and smart. You got that right. And of course, please pick up our primal edge, our one-shot wonder, our 310 organic liquid-cell-ready ingredients all there to make the healthy and get the good stuff in. And the bad stuff out. And if you're up early, you want to give us a call. It's<PHONE_NUMBER>. Good morning, Paige. Good morning. Well, this is our last show for the walk, because we're both going on vacation. Yes, we are. It's a beautiful thing. And I wanted to start off the program talking about food processing because that's always in the news. And there's been a lot of studies out. But I thought, first, what we do is just kind of go over what food processing is and what it means so we can define it so we know what we're talking about. And this comes from John Hopkins. Oh, there you go. Good source. Yeah, it's a good source. And food processing has come to be associated with, of course, ramen noodles, canned meat. Like spam. Yeah, kind of like those, what are those? Twinkies. Yeah, all kinds of stuff. Little devies, sponge cakes. Yeah. And businesses use food processing techniques to transform raw foods and ingredients to new products. These techniques are in the gamut and include milling grains into flour, probably one of the most processed ways of doing things, crushing seeds to get the oil, turning the milk into butter, mixing ingredients to make batter and baking cookies on the assembly line. Yeah, but some of these food processing techniques, although many of them might use cutting-edge technology, others have been practiced for millennia, literally, thousands of years, even before food scientists figured stuff out, how to make those little dinosaur crackers. It's been that long. Chicken McMollies. We know that fermenting and leavening bread, the Egyptians were leavening bread and brewing beer and all that kind of stuff. But for better or worse, Nico, all of our food is processed in some way, sadly. Yeah, and even meat processing can involve things like just cutting and deboning, which doesn't seem too bad, but also removing guts from the carcasses, which probably is healthy in a sense. So we have to think about this. Kind of the balance between a longer shelf life and actually some processing such as fermentation, which is a thing that we do to make the food more digestible. You and I were talking about that. Right? But some of it raises concerns over dietary health and safety in general. Yeah, especially because of that study that was out just a while ago that talked about ultra-processed foods that are much more dangerous for you. So preparation and food safety, over time, yeast, molds, bacteria and other forces cause food to spoil, making it less nutritious, of course, but it tastes bad and potentially unsafe. So many of the oldest food processing techniques, such as drying fruit, pickling vegetables, salting meat and fermenting dairy to make cheese and yogurt, remain important to this day because they preserve food and they delay the spoilage. Hey, listen to this. This is a little sniglet. Okay. In the 1790s, Napoleon Bonaparte offered a prize to the scientist who could best develop ways to preserve food for his armies in France. And Nicolas Appere, a chef from Paris, won the prize in 1810 for his discovery of sterile canning, a processing technique that is sometimes called appetization, and canning involves either glass jars, like the pretty colorful ones there, or metal cans. Interesting, when you go in a lot of grocery stores, like in Europe, you don't see so many canned metal cans. Mostly you see the jars, which I really like. Because you can see the stuff. Yeah. That's why they do that. I like it, too. Yeah, preservation allows people to ship foods over greater distances. They can store things longer, so you can enjoy them later. Processing can also inhibit or destroy pathogens, like disease-causing organisms. Yeah. They can also contaminate food. And you see that. You know, we've had a lot of food recalls and things that are going on. Right. So preservation techniques such as refrigeration, fermentation, dehydration, and the use of salt, sugar, or other chemical preservers can slow or stop the growth of these pathogens. Heat process such as pasteurization and cooking them can also help destroy them. Yeah, that's for sure. So although the businesses use food processing techniques to reduce these food safety risks, the facilities where food processes are being done sometimes are part of the problem, too. And large processing plants, for example, can handle large volumes of products from different sources. I think that's a problem. If you get a shipment of spinach, one farm, and that's contaminated with something, and then you're using the same equipment to do another type of vegetable or even process it. Cross-contamination against that. Yeah. Hey, here's a cool picture over here. I don't know. I don't want to click on it. Yeah, that is, you know, we're talking about some fermentation. Some processing is good for us. Some, not so much. But that was a picture of, it looked like stringy beans over there that I was showing you. And keep one more. There you go. Nato. That's natto. And that's the fermented soybean. You know, last show, we spent quite a bit of time talking about one of the most precious nutrients we can get is vitamin K2. K2 really protects our arteries and keeps us healthy. And usually only found in animal products. Much of it is. Yeah. Natto is one, but it's known for its pungent odor. That's why so many people like to take it. What's the process of fermentation that makes the K2? There's a precursor in there. It's made in our guts, you know? Exactly. But in any case, you know, I think that it's interesting to see that did we take processing maybe too far? Well, contrary to popular belief, the effects of food processing on nutrition are so varied that it cannot be generalized as unhealthy or even healthy. There's certain things that we did to food in the olden days to make it much healthier because somewhere along the line, we learned to do these things. Some of them were accidental. It was part of survival. Well, like that story from the samurai that left his rice out too long and he had the natto and discovered this was a healing thing and when he went into battle, he felt great. Yeah. All the other guys were going, You're eating that stinky stuff. Yeah. So, I mean, a lot of these things are accidental that we found, but essential to maybe our survival in the long run. And here's another thing to consider too. You know, here as we go down, I'm just putting up some pictures here. Here's a woman with a goiter and a lot of what we see as a result of processing. Maybe we remove some of the good stuff and put some of the bad stuff, you know, don't get the bad stuff out. Some foods have been processed. You know, when we started to realize that goiters were an iodine deficiency and that's when we started having iron fortified, I mean, iodine fortified salt and so forth and they added iodine to table salt to prevent the goiter. So, sometimes we're able to address health things by adding things that aren't there. That's right. Interesting. We'll talk a little bit more about processed food and we'll be right back after this short break. And pick up our health signals laser and please pick up our primal edge during the break. That's why we need primal edge daily nutrition. It includes a special blend of ionic, soil-based vitamins, minerals, fatty and amino acids in an easy to use liquid form. Primal edge is powered by highly concentrated folic and humic acids. Nature's preferred delivery system. They've been called miracle molecules because, like sunlight, air and water, life cannot exist without them. That's right, Paige. They ensure we receive all the nutrition we need to be healthy and thrive. We take it every morning. Primal Edge. Formulated and approved by Nico and Paige of Living a Primal Lifestyle. Buy it today for just $89. Click on the Primal Edge banner on the front page of TFNN.com. Using this first-of-its-kind program, the Art of Timing the Trade charts allows you to scan thousands of stocks for Fibonacci formation setups, including guardleafs, ABCs, butterflies and much more. The Art of Timing the Trade charts is designed to help you when scouring the markets for stocks just beginning to form the trading patterns that many investors spend days, weeks or even months searching to find. And right now, we're offering licenses available at only $79 a month. We are so confident that you're going to love this new charting software that will even give you a 30-day unconditional money-back guarantee. Don't miss out on this incredible new piece of software. 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And some processes deplete the nutrition, and some of them add it in. Sort of fortifying it when we fortify things. We add these vitamins in it. Can we just hold on for a second because we've got a question. Oh, okay. Zee wants to ask us, hey, we don't care when you tune in just as long as you tune in, no problem. I would ask you both to restate your brief conclusions on the benefits, hazards of plant-based and lab-grown meat protein substitutes hitting the market and being intensely promoted since I value your experience and knowledge. I would be very interested in your thoughts. Yeah, John, you know, that's a... Well, you know, we were... It's kind of what we're talking about in a sense. Right. Because really, you know, if you go back in history and look at what we ate as hunter and gatherers and people who are still hunter and gatherers mostly still eat this way. They're mostly talking about animals, insects, snakes and things like that. And eating vegetables as an afterthought. If we need them, we can use them. These days, the big agenda is for pushing for things that are not meat. I call it franken food. Yeah, you made a comment when we were on the break. Nico was sharing a documentary or a podcast that's going on. Yeah, this podcast came from down under. It's about three years ago, but very heavily on YouTube right now the healthiest diet in the world. And of course, what do they say the healthiest diet in the world is? It's all plant-based food. And there's arguments between them because it's very interesting. I was telling you during the break, one guy is arguing, well, you have to ferment these beans if you're going to use these beans because you can't use them in the other. And the other guy is saying, well, beans are very important. Let's skip that. It's so important that we eat beans. So they're all confused. And in our opinion and in our knowledge and our experience, what we know is that our ancestors passed down the wisdom that if you're going to eat a bean, you better soak it and you better soak it two and three and four times and change the water and then cook them to death. In order to protect your gut, we were not designed to eat these foods. We do not have three stomachs like a cow. This is why, over time, I actually think that perhaps then we've got the issue of juicing, you know, takes out the fiber so that the damage and a lot of the things that can happen for the plant-based side. Well, then you've got your sugars. Then you've got your sugars, you know, irritating the arteries. Remember, we talked about how important K2 is to protect our arterial health. So I'm getting back to what you're saying. These protein-meat substitutes, Nico and I have shared a thought idea. What was soy-based, most of them? Well, they're soy-based and you've heard us say soy is not a toy for the girl or the boy. This is the reason we see young girls that are having periods at nine and little boys that are kind of puffy, you know, a lot of these little vegan vegetarians. I see a lot of my friends on Facebook that are raising their kids and grandkids on, you know, healthy plant-based foods and you can see a difference in the way their kids look. I truly believe as a result of probably not getting the necessary animal nutrition. If you're on a plant-based diet for 10 years or more, you're going to have serious problems. I'm not saying that I love my plant foods, actually. I do, but it's almost like as a little condiment to my main protein source. And that's mainly how it was always used until the food companies started getting involved and pushing this on us and changing the way we did through the advertising, through the memes of breakfast. I've never had breakfast and I always tell the story of, you know, my first breakfast. They left when the sun rose to go get the work done before it got too hot. That's right. And then they came back. To answer your question, John, I really think that some of this push is, you know, you go back and hear the story about the kings, the certain foods were only allowed for the kings, or even when they said... They only killed the king's deer. Yeah, right. You can't do that. Only the king can.  Now some people might say, well, there's an access because, of course, nobody hunts anymore. Well, it's pretty well known that during the 1700s and we were just coming out of the Dark Ages and now we have this big revolution where we're starting to build machines and we all of a sudden stubble along a whole new continent, two new continents. So now we've taken the world and given half of it again to all the humans saying, you know, we're not running out because Europe was running out of stuff. Not so much Asia because Europe was running out because too many people there. We overfished.  So now we discover a new world and we go and ravage that. But I think what's really cool is like how you taught me when I first met you.  They wanted the royalty wanted the other foods so they could feed the masses on the rice and so forth. The food was scarce. So whenever I see them pushing things like this, I think it's to get the masses to accept these, I call them, subpar foods, franken foods. Well, first of all, much, much less expensive to keep food that you can process. Plant food, if it's fresh, goes bad fast, meat even faster. So fresh food, this is why you're coming up with substitutes because the fresh is hard on us, folks. Well, the other thing is that it's not enough profit. We were talking during the break. They had this thing in the paper of the CEOs in Florida and the ones they were talking about, four or five of them were making between $15 and $22 million a year and their employees were making $20,000 a year. That's the discrepancy I talk about in today's society. It's not equal anymore and when we start out on the land, we start out equal. It may not always be equal but we start out that way. Yeah, you and I have different opinions. I definitely don't want to be so equal that we think that socialism always leads to communism, in my opinion. And I'm not talking certainly about socialism. I just think you're exactly right. We have become convenience people and we are willing to accept this food and I'm going to say to you, John. Because we're brainwashed. Yeah, don't buy these frankenbrews. If it's highly processed, if it's a vegetarian protein, a little bit of it. I mean, I don't mind getting peas approaching when I have a peas, when I have them. But I like anything I have now fermented. I've noticed my stomach feels better. And I am highly suspect when they start promoting these other kinds of foods and these things that they know what's coming. And as David Dubine, we've always had Adapt 2030, he's been educating us on coming food shortages due to the grand solar minimum. If we have crop loss, all of you guys that are traders that are watching the show and tipping it in. You're just starting to see it now. We're seeing that due to bad weather and rains and cold and crop losses. And so now is the time to go buy that big bag of rice and store it in case you need it five years from now. So these things are valuable for me. And go buy your canned butter. Did you know they have canned butter? Yeah, canned butter. I've got so many cans. I've got a closet that's filled with cans. It's a beautiful thing. I mean, that's what you want it for.  And maybe that's exactly what we all need to do. I mean, in the reality. Remember, our show is based on local and seasonal. So if you buy local and buy seasonal and buy fresh food, you're going to be okay no matter what you do. If you eat more vegetables than I do, no problem. If you eat more meat than I do, good luck. It really doesn't make any difference as long as it's fresh and it's grown right to where you are. Bingo. That is our real premise here. And I've kind of gotten bored to tears on whether or not if you think a vegan diet is the best diet on the planet or you think of what your carnivory is. There are benefits. As long as you do them, there can be times where one might be the other. But I just don't want to argue that. What you need to have is, I think, a well-rounded diet with the foods that are fresh. Yeah. So in this thing, we're going to go over a little bit more of the processed foods. Yeah, we were saying sometimes processing adds good stuff to the food. Sometimes processing takes it out. For example, some processing techniques will actually add nutrients like B, iron, fiber, iodine. We talked about that. And in a case, a lot of cases we have to because the food doesn't have it. So second round, we'll have a lot more. We'll be right back. But the personal training studio that Niko is the owner and president of, Performance Training. Since 1998, Niko has trained individuals and groups to improve their health both mentally and physically. As a certified personal trainer, Niko's main focus is on demonstrating exercises correctly to avoid injury and teaching his clients how to manage their past injuries while getting the most out of their personal training sessions. The Performance Training Studio is filled with unique training equipment that enhances balanced results at a faster rate while minimizing damage and discomfort. For more information, you can give Niko a call at<PHONE_NUMBER> or email him at Niko at TFNN.com. Let him know you heard him on TFNN and save up to $100 on a special package just for TFNN listeners. Act today. The TAS Profile Scanner is the most revolutionary piece of trading software that you will ever try. Wouldn't you like to approach the markets with confidence? As you begin your trading day, it's likely that you'll be faced with lots of decisions. In order to make the best decision, the first thing you'll need is a strategy that will help you minimize your risks. Whether we're in a bull or bear market, a good strategy is to have the tools needed to help you scan and analyze the markets before you trade. The TAS Profile Scanner instantly scans and filters over 2,500 global financial markets such as stocks, ETFs, commodity futures and forex. 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See for yourself how you can harness the full power of the TAS Profile Scanner by visiting the front page of tfnn.com today and you'll find the TAS Profile Scanner under the Services section. Remember, with a 30-day money-back guarantee, you have nothing to lose. Don't let another day pass you by without trying out this amazing piece of software that will revolutionize how you look at the market and how you place trades. Sign up today. This segment is brought to you by Think or Swim. For more information, just click the Think or Swim banner on the front page. Welcome back. So when nutritionists recommend avoiding processed food, they often refer to highly processed foods that are high in added sugars and healthy fats and have little or no nutritional value and no longer resemble the plant or the animal that they were made from. Now you, during the break, you were telling me a little process that you make. This is a homemade processed food that would be very healthy. Well, yeah, for example, I think what it illustrates is me trying to think more like a primal woman, but yet with some of my little things that I like. So for example, I buy a chicken sometimes. In a hurry, I didn't roast my own chicken. I could roast my own chicken, but say I bought an organic roasted chicken and I want to have a chicken meal. So I also have some raw dairy. And maybe my raw cream is getting a little past the way I like it in my coffee. But remember, raw just ferments. It doesn't actually go bad. So I'll make like a chicken marsala mushroom with it. Pull the chicken off there and I'll make the chicken marsala. And then I take the bones and I make bone broth. Add a little apple cider vinegar to pull the minerals out. I cook that. And then after I strain the bone broth, this is kind of like a primal person would use all of it. I take, in the bones, I've cooked it for a day and a half. So the bones, I can literally smush the bones. So I add that, the bones, to my blender with a little bit of the bone broth. And I make like the... It's a little slurry type of broth. It's like a slurry and I pour it in ice cube trays. So basically the next time I have broth or I'm cooking a sauce or something, I'll add one of those and then I'm getting bone marrow and all that delicious, healthy collagen and so forth. And then I take my bone broth, I put it in my big thermos and add a little pat of butter and some pink salt and it's delicious. And then you and I talked about eating the organ, eating the whole animal and that's the way we were raised to do. And if you don't have organ meat, because most people don't see it, you have to really ask for it at the grocery store. We're going to talk about a company that offers organ meats in a capsule that are processed but freeze dried to preserve the nutrients. And my recommendation is to take maybe 10 of them capsules once a week, maybe twice. And that would be more like you would have, you wouldn't be eating organ meats every day. If you were primal and you had a hunt, you might have an animal and then after that was gone, you lived on the bones and the marrow. So you shared with the whole tribe. Sure. So you made up a soup to make it last. Exactly. I like that. These are things that are very easy to do. And I love the idea of the ice cube thing. Then you can plop it in whenever you need it. And the next time you make a soup, it's a more nutritious soup, right? Yeah. I just want to share that. So now we're talking about, on this about animal slaughter, meat processing and the workers' health. And that's the key point. A lot of people don't realize, we've seen some of those awful documentaries that show the horrible conditions of that type of processing plan. And you know, what's the state of the meat that people are buying? That's why we're very picky about where we do buy it. And there's a huge difference between going into a factory, even a small factory, and watching them do that, which if you can get in to do that, a lot of companies will restrict that. Or going to a place where this one guy is a family farm and they're processing the meat outside and they're glad to have you watch them.  It's very humane. Remember animals are usually in the wild, eaten while they're living. So this is a much more humane. They're killed very humanely, very quickly. It doesn't cause any pain or distress to the animal. In fact, that's what we don't want. We don't want stress. We don't want, when you're hunting, you don't want the animal to see you and you don't want to be chasing it. You want an animal that's at peace with itself. But I think what I get from this, Nico, is that we've gotten so far away from being involved in our food. And I think there's a lot of spiritual benefit to being more concerned and involved in where your food's coming from and comparing your food. But this is a society problem because not only are we not too concerned and not involved with finding and processing our own food, but we're not even doing that to our own children.  We send them off to strangers as soon as we can. And then they're educated by strangers. Maybe not the same values as you. This is something that our primal people would never do. We never would give away our children. Our children need to be beside us so we can show them what life is about. Yeah, it takes a village. Right, it takes a village. Remember that saying? So it's the whole process of thinking how we do things today is completely disconnected with how we evolve. And that's the premise for the whole show. Showing you how we can now add things in to be more primal, to be healthier, to have a clearer mind, and not to be a slave to our modern society. No matter if you're a communist or socialist or capitalist, it doesn't make any difference. All these isms are the same thing. Just the sky's a little bit different. And believe me, I love living in the United States. I would not live in any place else. Hey, you're exactly right. I want to thank you, John. Thank you for the nice feedback on our show and for sharing our information. The idea is this is news you can use to stay healthy. So thank you for that. And I mentioned the ancestral supplements. Remember, I mentioned that perhaps we don't get the organ-based nutrition that we were supposed to. I thought real quick, maybe we can kind of talk a little bit about this way of... This is a type of processing that actually... Yeah, we can interject, but I think this is a good example. Here's a company called Ancestral Supplements. And they produce one small batch at a time to ensure higher quality and attention to detail. This includes third-party testing of their products. And they're a small Texas-based company doing business the old-fashioned way. No shortcuts, higher quality. Just what Nico was saying, higher standards where the customer comes first. And they basically have one purpose. I'll have what he's having. Exactly, that's what I thought you'd say. He looks good. So basically, his mission is... I'm all distracted now. He's putting back in what the modern world has left out. And that is to return people back to strength, health and happiness. And ideally, we would consume the whole animal, nose to tail, to optimally support and sustain a vibrant disease-free life. And for most, that's just not practical today in modern times. That's where all this has come in. So their supplements are for individuals seeking fundamentally liver, bone marrow and organs and targeted support in harmony with nature, the old-fashioned way, the way our early ancestors did. Yeah, his name is Brian Johnson. He's also known as the liver king. And he believes that the evolutionary theory of health and wellness, he believes in that theory. He believes that for millions of years, robust strength and exceptional health was the center to our existence. We once were strong, lean, healthy and fit. I believe this because most of our human history, we effortlessly consume, nose to tail, all the things we needed for health and happiness. Like the fertile ground that we once walked upon, we are a natural extension of this earth. And then here's the key point, and I know you guys have heard me say this over and over.  And because of this, they decided to start a supplement line that makes the biggest impact, that gets some of these key nutritional components in with the highest quality, grass-fed beef liver and bone marrow. We'll put a link. I actually enjoy these products and have them delivered. And as I said, the way I do it is, you know, I'm over that taking this and that, you know, and you've got all these pills. You take them almost like a meal. Or you can open them up and sprinkle them in your food. Yeah, so that's another way to do it. We have to get out of the break. We'll be right back. But today, our food sources no longer contain the vitamins, minerals and nutrients our bodies need to stay healthy and strong. That's why we need primal-edge daily nutrition. It includes a special blend of ionic, soil-based vitamins, minerals, fatty and amino acids, and an easy-to-use liquid form. Primal-edge is powered by highly concentrated folic and humic acids, nature's preferred delivery system. They have been called miracle molecules because, like sunlight, air and water, life cannot exist without them. That's right, Paige. We receive all the nutrition we need to be healthy and thrive. We take it every morning. Primal-edge, formulated and approved by Nico & Page of Living a Primal Lifestyle. Buy it today for just $89. Click on the Primal-edge banner on the front page of tfnn.com. It's amazing to think that Tom O'Brien started his weekly gold report 17 years ago with the first issue published April 7, 2002, when gold was trading at under $300 per ounce. Gold peaked at more than $1,900 in 2011, and after spending many years consolidating at lower prices, gold may be poised for its next big run. Tom O'Brien publishes his weekly gold report every Monday morning for subscribers, consisting of coverage of the XAU, HUI, GDX, the Dollar, Bonds, South African Rand, as well as 25 different mining equities with specific buy-sell recommendations. As of April 1 of this year, the gold report currently has eight active positions with an average unrealized profit of almost 8% for each open trade. New subscribers get a 30-day money-back guarantee so you have nothing to risk. For all the details and to start your gold report subscription today, visit the front page of tfnn.com.  Sign up today. 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Get in on the ground floor of the next big thing today. Don't forget, you can listen to tfnn live on your mobile device 24 hours per day. Go to tfnn.com and hit Watch Tiger TV. That's tfnn.com and hit Watch Tiger TV for the latest market information. And welcome back. We're on a site called Ancestrialsupplements.com. This will, of course, be in the next newsletter coming up. And his mission is putting back in what the modern world has left out to return people back to strength, health, and happiness. Because, fortunately, our potential for strength and health remains encoded in our DNA. And under certain conditions, we can genetically express the strongest, healthiest version of ourselves. And those certain conditions almost always have to do with nutrition. It's the first thing that matters. And I was talking to Ellen last night. We were just kind of bantering before going to bed. And I was talking to her about having the food companies be responsible for our health. Wouldn't that be a novel idea? Yeah, they pay for all our health coverage and everything like that. And now the alphabet soup organization in our government is responsible for policing these food companies and so forth. Well, you know, they take money to pass things that aren't healthy. Well, not only that, we have to revolve the revolving door of going from CEO to the government, back to the CEO, back to the government. And so they're not really, it's like having the fox guard the hen house. That's right. So if you are like the Johnson tribe or family, then you eat nose to tail diet.  Hey, fish sauce, for example, when I went to Thailand, that is a staple in their diet. So people go, oh, the people in Thailand, they eat a lot of rice and noodles and foods that came from animals. And that's what makes the big difference there. So this is really cool, folks. And this is all available through Amazon Prime. So it's very easy to get. Look at this. Grass-fed beef heart, beef thyroid, pancreas, spleen. This is where food is thy medicine, folks. Bone marrow, thymus. And these are easily to take. You can sprinkle them on your food. You can take them as the capsule itself. But this is all the pure product. This is free-dried, very healthy. And this is what I think you should be doing. It's not our product, but I'll push this to the limit for sure. So we'll be in the next newsletter. Yeah, and I suggest that people really get going with it. A lot of benefits in this. When he's talking about the products and everything like that, he's talking about feeling vibrant. He's talking about sleeping like a rock. He's talking about waking up with a smile on his face.  See, he's got it right. This guy's all about living a primal lifestyle, eating the whole animal, being in tune with nature, activating the ancient pathway of sleep, nutrition, movement, and sun. Yeah, I'm just saying, he's a strong man. He's a man's man.  But, you know, like what he says, he has to make his little disclaimer, I'm not a doctor and this is not medical advice, but having a degree of biochemistry knowledge, he wrote a thesis on the chemistry and biochemistry of supplemental nutrition. And as a, he's also a primal blueprint certified coach, you know, to Mark's thing. And he's been obsessed with ancestral health and living for over 20 years, and he cares about people, and I really like that. So one of his, you know, he doesn't just share his supplement, he's also sharing tips and tools, and tools, which you and I really hold dear, and that's getting in touch with nature. And one of the most important things we can do, and we talk about it a lot, is really getting our sleep hacked.  And it's really sleep triumphs almost everything. And that's a big problem today for everybody, it seems like. Well, and then let's tie it back to, is it this highly processed food that they average, you know, the franken food? Well, that is not eating enough sun, so he asked the question. He says, how badly do you want this? How badly do you want to be strong, healthy, happy again? What are you willing to do? If you're like most people reading this right now, there's a little that you wouldn't do to retain or to reclaim these virtues.  And that's definitely not the way to go. Yeah, but, you know, do we want to share his, you know, if you read his tips of what he says, listen to this, I want to go over these because people like these tips. I like them. Hey, who does this sound like? Get morning sun exposure at the same time every day to regulate your sleep and wake cycles. In other words, establish a circadian rhythm. See what he says, this is how you regulate yourself by going with the sun. As I said, our eyes are not just cameras, they're clocks, so set your clock and everybody knows what to do. Finish your last meal at least three to four hours before bedtime. If we are eating before bed, then our body cannot rest. It has to digest. And that digestive system needs a rest, too. Yes, and my wife being home, working from home now, 99% of the time, allows us to eat at four o'clock instead of five or six, which is a huge thing digestive wise because I go to bed at nine o'clock. And when you go to bed at nine o'clock, there's hormone stimulation between ten and two. You're repairing and you're getting younger as you sleep. Yeah, I like that idea. The other idea is get grounded. Get out the outdoors, connect with the magnetic surface of the earth. You can do earthy, you can get grounding sheets so you're grounded. Or you can go on barefoot hikes on the, you know, on the ground and go to the beach where we live. And put your phone in airplane mode to make sure there's no electronics, no lights anywhere near you when you're sleeping. This one, do you see what it says? Your sleep temperature should be between 64 and 68. And boy, I like that. I cranked my air down and, you know, our air-conditioned guy, you know, it's working fine because I do it right before I'm going to bed. Yeah, ours clicks down to 72, but then we have a nice hand. Oh, that's hot for me.  Your body's actually take that as a signal that it's cold and dark and it's time to heat up the repair engines. And use your blackout curtains and blackout blinds so that you have that dark. Again, it's part of that whole signaling. How about this? Use Himalayan salt lamps at sunset instead of regular lighting. That's what you do and we do. And I've gone in all of my rooms that I transit at night. I have a lamp and I bought the red incandescent light bulbs on Amazon. And I've replaced a lamp in each room that is just the red light. So I can... I don't have Himalayan salt lamps in the whole house, but it would work. But I think this actually provides a little better light and it's definitely a red tone. The other thing the author does, he recommends, of course, a ketosis type diet, ancestral type diet. And that is really a diet that calms everything down. Not only the body, but also the mind and everything else. So he really goes deep into that ketosis and a high fat diet is the thing to do. Right, and obviously having the whole animal and getting all the nutrient-dense foods, which are really animal foods, but he also recommends the blue blocker glasses at least two hours before you go to bed. That helps make sure that your eyes aren't being stimulated at the wrong time with blue light. And this is my other favorite one that he says.     Most of us, when we relax, and our head goes back, our jaw relaxes, and that obstructs airway. When your airway is obstructed, you don't get the oxygen to the brain that you need to. And it's a game changer. You end up feeling very relaxed and you sleep much deeper. Last but not least, he says sleep on the floor. Like the tribes did. Use that magnetic connection. Maybe that's a good idea. I think that's probably a good idea for sure. Make the whole bedroom of floor put mattress in the whole thing. Just like a futon. It's like around folks, we've got a little pour. We'll be right back. When it comes to managing your money, let me teach you to do what most wealth managers tell you can't be done, which is how to time the markets. I'm Steve Rhodes, author of Mastering Probability and for the last 12 months, Timer Digest has been tracking my newsletter signals which have earned me the ranking as their number one market timer in the nation for the S&P 500 for the last 12, 6, and 3 months. Timer Digest also ranks me as the number one market timer for gold as well. The fact is markets can be timed and I'll teach you the exact set of tools that I use that have transformed me into one of the best at what I do. Sign up for Mastering Probability today by clicking on the newsletter tab on the homepage of TFNN.com and get immediate access to workshops where I take you step by step how to use an extraordinary set of tools as well as provide great market calls too. Sign up today. Hi folks, Tom O'Brien here. If you'd like to get my daily newsletter of Market Insights, then now is a great time for a 30-day free trial. Every morning by 9.30 I send out my morning letter to subscribers with Market Commentary on a variety of markets, currencies, and commodities to keep investors up-to-date on the day's trading action. Included in Market Insights are specific buy and sell recommendations for stocks, ETFs, and even options, which stops and price targets included for every trade in my newsletter. If you'd like to try my newsletter risk-free for 30 days, then head over to the front page of TFNN and you'll find Market Insights under Market Insights. I use my years of trading experience to bisect and dissect the market every morning and give my subscribers the most important information they need to know for the day ahead. I even issue afternoon updates for my subscribers whenever warranted with important market action. I'm always scouring the market for the next great trading opportunity. Sign up for your 30-day free trial to my daily newsletter Market Insights today by visiting the front page of TFNN.com. We have a lineup of live content for traders by traders every market day, featuring some of the most knowledgeable and respected minds in the business. TFNN broadcasts five days a week live from 9 a.m. until 5 p.m. Eastern Time. We have live programming every market day during market hours. Every morning Larry Pezzavento kicks off the trading day live at 9 a.m. and breaks down the opening bell with trade what you see. At 10 a.m. Tom and Tommy O'Brien host the Tiger Technicians Hour, followed at 11 a.m. by the team at TD Ameritrade and Thinker Swim with Fast Market. Basil Chapman host the Tiger Technicians Hour at noon, Steve Rhodes at 1 p.m. with the Traders Edge, Dave White at 2 p.m. with the Power Trading Hour and Tom O'Brien anchors the daily lineup from 3 till 5 as host of the Tom O'Brien show. Tune in to TFNN's Tiger TV on your computer or mobile device and you can always find us streaming on YouTube. TFNN.com Welcome back to the last segment. He talks about here, he says, I follow a seasonal eating patterns that allow us to slip in and out of ketosis. If we're feeling really good, we may stay there a little longer. It helps us to think, to perform and to produce. The following is an idea of how and what to eat. Notice that we also go through periods of feast and famine within a 24 to 120 hour fast and huge refeeds where almost nothing resembles primal food is off-limits. So he talks about water fasting, consuming only water and minerals. It's like the single most powerful metabolic intervention in existence. Well, second to a dry fast. Now, dry fast is actually an amazing bet. It's accessible to everybody. It's time and money saving. It's a default way of life in the animal kingdom. Only recently have certain animals, homo sapiens included, had such an easy unlimited and effortless time to find food. Well, think about it. Let's go with our whole discussion of this show. That fasting is very much what a primal person would have experienced. They're looking for their next meal and it may not just be like what we have where it's every two hours at the nearest wah-wah. They pull in and grab something. We haven't got it completely backwards. So I like following people like this. And certainly the products are fantastic. And this is, of course, going into the health signals newsletters, they can read all about it and have access to these sites. But folks, it's really important to understand that highly processed food is not the way to go to each seasonally local fresh as you can. I don't care if you're eating animals or plants, but for me, the animals are the staple that revolves around 90% of what I eat. And the rest of it is not as important. It's more important for me to get what I need from the animal and if I can include some of these things that Brian Johnson is bringing to us, that really, really feeds my brain and the whole body just feels really good when I'm eating this way.  And we didn't even finish it. And that's all we had. We had a couple pieces for breakfast. I have two eggs and a little sausage. I mean, there's not much. You don't need much. They've been fooling us. That's exactly right because they sell a lot of that junk that's in the shelves. This is great, guys. Get this article, read about the tribal bonding. It's a good little section as the show. Not bad. We'll see you all after vacation, I guess, huh? Yeah. Bye-bye. Thanks a lot.


You and two friends order a pizza and it's been cut into 10 equal slices. One of your friends grabs five slices, the other grabs three slices, and you're left with two slices. Now you're trying to figure out who got the least amount of pizza. In this scenario, you probably easily realize that you got the least amount of pizza, having only gotten two slices. You're right, but how did you know this? Because the slices are exactly the same size, the one who got the fewest slices got the least amount of pizza, and the one who got the most slices got the most pizza. Now how does this talk of pizzas relate to fractions? Fractions are basically portions. Since the pizza was cut into 10 slices, each slice was one tenth of a pizza. The one who got five slices got five tenths of the pizza. The one who got three slices got three tenths, and you got two tenths. Just as you were easily able to compare pizza portions, because the size of the pizza slices are the same, it's easy to compare fractions that have the same denominator. When the fractions have the same denominator, simply order the fractions according to their numerator. Take for example the fractions five tenths, two tenths, and three tenths. They all have the same denominator, ten. Since you know that two is smaller than three, and three is smaller than five, you can arrange your fractions as two tenths, three tenths, and five tenths. Two tenths is the smallest fraction, followed by three tenths, and five tenths is the largest fraction.  What if some of the slices are bigger than others? You wouldn't be able to just count slices as before. Let's return to your pizza party. Some slices are smaller, and some are bigger. One friend gets three of the smaller slices, another gets one smaller slice, and you get three of the bigger slices. You notice that the bigger slices are twice as big as the smaller one. So, if you cut the bigger slices in two, you see that each big slice is the same as two of the smaller slices. Your three big slices are actually the same as six small slices. Just as you cut the bigger slices to look like the smaller slices, so you can compare them, you have to change the fractions when you compare fractions of different denominators. The trick is to change your fractions so that all the denominators are the same. By doing this, you can then compare the numerators like you did earlier. Take for example, the fractions three tenths, one tenth, and three fifths. The first thing to do is to find the least common denominator. The LCD here is ten. Since ten is both a multiple of five and ten, the fractions then become three tenths, one tenth, and six tenths. Since the fractions now have the same denominator, we can order them just like we did earlier. One tenth is the smallest fraction, followed by three tenths, and six tenths is the largest fraction. There you have it. Even though you got three slices, you actually got six tenths of the pizza. Let's try another example to see if you understood it. Order these fractions, four fifths, three quarters, and five tenths. The lowest common denominator is twenty. So we rewrite our fractions as sixteen over twenty, fifteen over twenty, and ten over twenty. Since ten is less than fifteen and fifteen is less than sixteen, our fractions in order from smallest to biggest is ten over twenty, fifteen over twenty, and sixteen over twenty. There you have it. If you liked the video, give it a thumbs up, and don't forget to subscribe, comment below if you have any questions. Why not check out our Fuse school app as well? Until next time.


The prevalence of mutated strains of SARS-CoV-2 has created a need for a cost-effective, rapid and reliable method of detecting virus levels in public spaces. Researchers are exploring new strategies to target, bind, and confirm the presence of viruses before they spread. One approach involves synthetic DNA constructs designed to specifically recognize SARS-CoV-2 antigens. These constructs can then be used to create complementary DNA, cDNA, or protein DNA complexes, which can be detected using small molecules, cDNA, or protein DNA complexes. Nanomaterials with unique electronic properties, like graphene, may also be used to develop a device with extremely low detection limits, septimolar or atomolar. To ensure accuracy, surface modifications must be made to the device to provide high sensitivity while maintaining rapid signaling capabilities. This article was authored by Narendra Chandrasekhar, Ramachandran Balaji, Ramaswamy Sundeep Parila, and others.


Savas is the perfect example of why finding a really good PhD student is both a blessing and a curse. It's a curse because he comes to my office three times a day and normally he doesn't leave unless I kick him out. That's also a blessing because what he wants to talk about is really really high level, very interesting scientific stuff, which is a very welcome distraction when you're waiting yet another boring report that no one will ever read. He's originally a civil engineer, believe it or not, and then he decided to take our masters, which is where we met. He took our module, my module, and was asking questions all the time, so I can't even say that I wasn't pre-worn, but he must have liked it because he then decided to do a PhD with us, which is doing on the application of our imaging methods to tissue engineering, which is expected to play a very important role in the future of medicine. So during the project, not only has he developed and honed his physics and imaging skills to a very high degree, but he's also developed this amazing ability to interact with the medical community and you don't need me to tell you how important that is for the sort of research that we do in the department. So if you want to take him along to your next meeting with the medics, maybe you can borrow him from us for a very small fee. And with that, I'll hand over to Savas and his new medical, new imaging applications for tissue engineering. Thank you, Sandra, very much. I mean, you oversold me a bit, so anything I present now is not going to be adequate up to the standard that people are expecting in the audience, unfortunately. I'm going to try my best, though. So today I'm going to present to you one of the chapters of my PhD thesis, which titles guiding the tissue engineering of Anisophagus using X-rays. This is a multi-disciplinary project which is supervised by Professor Sandro Alivo and Professor Paolo, the copy in the Institute of Child Health. So the case of isophageal atucia is when the isophagus goes to a planet that is still being connected to the stomach. So visualize it as if there is a gap along the length of the isophagus and if that gap is short enough then the clinicians can actually pull the two sides together and rejoin the isophagus. However, if this gap is long there is no current therapeutic option. The possible solution to this problem is to have a tissue engineer construct which can be transplanted in the isophagus and therefore bridge the gap between the two sides of the isophagus. The general requirements in order to have a tissue engineer construct are cells, the scaffolds of interest, growth factors and then you combine them together, you put them in incubation in a controlled environment and you have your final product. The work in ICH is currently in in vitro stages and we are working with piglets. So we have the native samples extracted from a piglet which is then desaleralized so it's stripped out of its cells in order to have the scaffold sample. Then the appropriate cells found in the isophagus are seated in the scaffold. It's put in incubation for 10 days and we have the final resaleralized in vitro matured organ. And as a final step, before imaging we critically point right in our samples. Of course the current gold standard process in the construct is histology. However, as you know histology is destructive, it's limited to a single orientation of dissection and it can actually go really long with the waxing and if you get the wrong, you lose information. It doesn't allow of the extraction of non-destructive volumetric microarchitectural information of the sample. And at the same time it doesn't allow quantitative intraneter sample statistics. So comparison. A potential candidate is X-ray tomography and I'm pretty sure you're familiar with a conventional attenuation based X-rays which relies on the degree at which X's are attenuated in order to create contrast between different tissue in a sample. This is an isophagus by the way tomography lies. However, as you can see and as you already most probably know X-rays suffer from soft tissue contrast. However, if we exploit the fact that X-rays are also phase shifted as it travels through a material and we have a device that can actually capture a shift, we can have soft tissue contrast and at the same time we can have a density map of the material we are imaging. Having a closer look, the phase based image compared to attenuation based you can see maybe not so well on the screen but we can distinguish the four isophageal layers of interest in the image on the left and it's quite challenging on the attenuation based image. And then this is the system developed by Sandro and it sits on the second floor of medical physics called the edge illumination system. Of course, all of the findings that we have with X-rays need to be confirmed with a cutting gold standard. So this is an example of a native sample and that we're actually image first and then cut it out histology, H and E which is the standard histology that the clinicians are currently carrying out. As you can see, all the features that can be identified in the histology are also visible in the X-ray sample and as this is a density map of the actual sample, regions that are cell rich like the muscle layer of the sample are going to appear bright, which means dense and empty regions like the lumen of the sofas are of course going to appear as dark. An example of a scaffold sample, which is cell free and the intensity of the walls now are significantly reduced because now it's desidualized so it's straight out of the cells. And then this is a case of a desidualized sample which I put in a parenthesis half seeded because at this stage the clinicians seeded cells in on the one side of the sample only. This is immunostaining an H and E on the same sample and you can see that this region is populated with cells and it appears bright on the X-rays and then the regions that appear less dense on the X-rays are the ones that have no cells. An example of fully seeded sample where it's evident when comparing with the previous one that this one is fully populated with cells along with a corresponding histology. So what can we do with our images? As I mentioned before histology suffers from a single dissection orientation. In our case you can actually use the different images stack them together and have volumetric information independent of the orientation. So we can extract information about the sample in all plates. Furthermore we can have a quantitative map of the different layers found in the esophagus and in this case following N127 from the outside towards the lumen of the sample we can see the serosa which is a connective tissue then we have a blood vessel which appears way less dense because it's empty. We have the two muscle cells of the esophagus. We have a very narrow peak which is basically a connective tissue. I'm not quite sure if you can see on the screen but it appears as a bright peak because connective tissue is dense. We have the loose connective tissue called the submucosa and of course the final layer which is the mucosa and epithelium which is bright and dense. In the case of the scaffold material again we can distinguish all the different layers but for this slide I want to highlight that it is very important that we can confirm that volumetrically all the different layers of the samples are preserved post-desertalization. So after we strip out the cells of the sample. This is a deserterized sample and the first thing that strikes your eye is of course that the entire microarchitecture of the sample is lost. We can observe ingredients from R3 to R4 of the cell population so we can confirm from the images that more cells for some reason tend to populate towards the inner side of the sample and that this is explained due to the fact that during the tissue engineering process there is flow of media through the lumen and therefore cells are moving towards the lumen and most importantly the clinician said that it's crucial to know whether this deserterization approach took place uniformly across the volume of the sample and I think that's something we can derive from our volumetric images. Of course to have more detailed statistical analysis on the samples we need to mask out the different samples so isolate from the surrounding. In order to do that we have option number one to do it manually which is going to take ages because I have over 50 samples and over 4,000 slices to do manually or choose the smart option of using a semi-automated approach developed by Alessia Azzeni which is a fellow CUT student and the only thing you need to do is basically input the first segmentation image that you've done manually the last one and then everything in between happens for you automatically. As I mentioned before the actual image is a density map of the sample and if we use the tissue mask that I showed you before we can isolate the sample and then for each slice we can have the average density value of the sample. Of course we can plot the density variation along the volume of the sample and for this example I'm demonstrating a native sample, a resaloralized sample, a scaffold. Both the native and resaloralized samples have higher density values than the scaffold and then this makes sense because these two are cell-rich while this one is just a resaloral matrix with no cells. However we realize that as you go across the volume of the resaloralized sample the density value drops the value of a scaffold. So this raised the flag and we checked out the images at the start of the volume and the end of the volume and as you can see visually it's pretty evident that the last slice has way less cells than the first slice. Then we thought that's the clinicians and the clinicians confirmed to us that that end of the sample was the part that was clamped to the bioreactor and therefore had no cell injections close by and therefore no cells populating the sample. Of course we can do the same thing for all samples but if we want to have a statistical comparison between 50 or more samples we need to express each sample as its own volumetric density value. I think I know everybody knows where this is going. We put them together in different groups and we have statistical testing on the samples. We have the native, the scaffold and the resaloralized and both the native and the resaloralized show significant difference when compared to the scaffold. And then this is some great news because you expect the resaloralized sample to have higher density overall than the scaffold and similar density values to the native. So every time we actually extract the different sample from the biological testing we input in this population here and we see where it's actually going to be sitting in the distribution. Another type of analysis that we can do is to compare the physical size of the samples and of course to do so we need to have a normalized value of each sample because okay it's fair enough to say that all the samples are derived from the same peak type but there is some variation because it naturally occurs. Therefore we decided to express each sample as a isophageal tissue percentage. So how much of the sample is occupied by the tissue? This is a case of three different types of samples. Apologies. This is a case from the same sample at three different stages. So when it was native you had the 70% tissue percentage, a scaffold it became 45% tissue percentage. So that's an indication that the walls of the samples got smaller because now they are cell free and then upon repopulation the tissue percentage increased because now we have populated sample. Again, statistical analysis. Both native and decentralized sample show significant difference when compared to scaffold. At this point, yes, this is indeed a confirmation that the decentralized sample indicates that it's populated with cell because the walls are increasing size but at the same time this indication of the mechanisms at which the scaffold is repopulated. I'm going to explain to you what I mean with this diagram. So have a scaffold then we have some cells injected in the walls of the scaffold and then the cells if I may they have the free will to move outwards or inwards towards the lumen. If we have seven an increase in the percentage which obviously is in decreasing the lumen percentage it implies that the cells tend to move towards the lumen of the sample and then we have the final tissue engineer construct. This was discussed with the clinicians and it was confirmed that it actually makes sense because cells tend to respond to external stimuli and shear stresses and as I mentioned before if we have the flow of growth factors in the center in the lumen of the sample then the cells tend to move towards those shear stresses. Of course they grow and proliferate more at areas where they can find growth factors and at the same time the sample the cells tend to favor regions that they have low density so the cells have the option of either moving towards a connective tissue called the xerosa on the outside of the sample or move towards the empty space which is the lumen and of course they rather move towards the lumen. To conclude we are able to demonstrate that we can integrate our exit technique with the current gold standard technology. We can extract volumetric information which can be vital for the assessment of the tissue engineer construct and scaffolds. We can have visualization of the samples microarchitecture. We can carry out quantitative intra-inter sample assessment and of course we can get an insight in the mechanisms of repopulation. I would like to thank everyone from the Institute of Child Health focusing on this guy Maria Ghirli which is the postdoc that runs the show if I may and from the advanced x-ray image group I would like to thank literally everyone because we're a team where everybody actually it's there and helps everyone else.


Hello, my name is Nina Kodra Eugene, Chief Transport Officer attached with the Department of Infrastructure, Ports and Transport. I wish you a happy public service day in keeping with the theme Embracing Digital Government Post-Pandemic. The Division of Transport, like many other organizations, was significantly impacted by the COVID-19 pandemic. The Division experienced a total shutdown during the months of February and March 2020, and this impacted the delivery of services to our valued customers. It was then that the drive came about to embrace the concept of digital transformation. Having recognized the need to better engage and serve our customers, the pandemic encouraged the acceleration of this process. The Division of Transport in partnership with the Division of Public Sector Modernization was able to respond to the pandemic in creating access to driver's license-related services online. This increased the ease of doing business. The new DigiGov platform went live on the 20th of June 2020, thus allowing customers access to the driver's license services from anywhere in the world. To date, there are 17 driver's license-related services online. The customer cannot only access the services online but also make payments via the website. These services can be accessed via our online portal at digigov.govt.lc. Some of those application services that can be found on the website would include change of driver's license information, application for renewal of driver's license, application for renewal of driver's license requests for exemption, application for replacement of driver's license, application for issue of provisional driver's license, application for learner's permit, application for renewal of learner's permits, application for omnibus learner's permit, registration for omnibus theory and practical test as well as the practical test for various other vehicle model types. The theme for Public Service Day, Embracing Digital Government Post-Pandemic, is a theme apt to this time period. The pandemic has taught us to embrace this new normal where life does not stop but adopt a new knowledge and methods of delivering services. The division of transport to date has processed over 55,000 driver's license-related services with 26% of these being complete online transactions. The services offered online have remained uninterrupted throughout the pandemic since its launch. Transport is currently in the advanced stages of the deployment of the vehicle-related services, thus substantiating 75% of our total services. With the current move to digitize, we have also been able to decentralize our online services providing access via the I-City Centers and the Castries SBA office. The division of transport wishes to encourage customers to use the online platform as opposed to waiting in line in keeping with the theme, Embracing Digital Government Post-Pandemic. I thank you.


In today's lecture we will be discussing Boolean functions we have already discussed in previous lectures Boolean algebras but still to start with I will recall the definition of a Boolean algebra we know that a Boolean algebra is a lattice which is complemented distributive having at least two elements and also having a least element and the greatest element apart from that it has three basic operations one called disjunction or or or plus and the other being the conjunction the and or product and the complementation operation now the probably this simplest Boolean algebra consists of the set 0 1 with the operations defined as this one which is the or disjunction or simply we may call it plus and then another operation which is called and or conjunction which is denoted usually by a simply a dot or by juxtaposition of two elements which is defined as this and complementation which is simply this usually denoted by a prime and 0 prime is 1 and 1 prime is 0 sometimes it is also called not now as we have seen before that a Boolean function this Boolean algebra consisting of only 0 1 can be extended to b to the power n which is the Cartesian product of n copies of b the operations disjunction conjunction and complementation or or and not are defined as this or or plus consider two elements x bar and y bar inside bn therefore x bar x bar is of this type x 1 x 2 dot dot dot xn belonging to bn and y bar is y 1 y 2 dot dot dot yn belonging to bn x bar plus y bar is x 1 plus y 1, and so on up to xn plus yn now this plus operation is same as the plus operation that we have defined before over b similarly I can define and or just the dot as x bar and y bar equal to x 1 dot y 1 and so on up to xn yn not that is the complementation denoted by the simple symbol prime is x bar prime equal to x 1 up to xn prime which is again x 1 prime dot dot dot xn prime it is fairly easy to check that bn along with the operations just defined which are induced from the operations defined on b is a Boolean algebra for any positive integer value of n. So if we are very strict about that then we can write the Boolean algebraic system involving bn as bn then this plus the dot the prime and of course the least element which is the all 0 vector having n terms and the greatest element which is the all 1 vector having n terms combining all these things we have the Boolean algebra bn in the beginning it seems that this is a very restricted class of Boolean algebras but we can show that any finite Boolean algebra is essentially same as bn we are not going into a theoretical proof of that or a theoretical definition of what is called isomorphism of Boolean algebras. But we shall write the statement over here that any finite Boolean algebra is essentially same as bn for some positive integer value n we want to define functions over Boolean algebras the first type of functions that we define and which are called Boolean functions are functions from bn to b a function from bn to b is said to be Boolean function that is what we are going to study today now let us start with examples of Boolean functions to start with Boolean functions are very simple. So let us consider possibly one of the most simple examples of Boolean functions here we consider n to be equal to 2 then we can enumerate very easily all the points of bn that is b2 those are 0 0 0 1 1 0 and 1 1 we can list them in a table and since we are interested in a function from b2 to b we can write a rightmost column in the table some values from b so it can be just 1 0 1 0. So suppose this column I am designating by the function f which is essentially the function that we are defining from b2 to b we can say that this function takes the value 1 when the input is 0 0 it takes the value 0 when the input is 0 1 it takes the value 1 when the input is 1 0 and it takes the value 0 when the input is 1 1 at this point we define something else we define what we call a Boolean variable so a symbol x is called a Boolean variable if it can take the values 0 or 1 so we are essentially defining variables what they said b a symbol x is said to be a Boolean variable if x can take values from b which is essentially the set containing 0 and 1 now we look again at the table that we have already discussed in the context of the function f the each coordinate point of b2 can be assigned to a Boolean variable so we write the same table like this we assign the Boolean variable x1 to the leftmost coordinate and x2 to the rightmost coordinate so we get the values like this 0 0 0 1 1 0 and 1 1 and now the function f can be thought of as a function on x1 and x2 and its values are 0 1 1 0 a table like this for a Boolean function is said to be its truth table it is of course obvious that if we have a function from b to the power n to b for any positive integer value of n then we will be able to write the truth table of that function at this point it is worthwhile to check the ordering in which we write the values of the vectors of bn so we look at b3 suppose x1 x2 x3 are the Boolean variable corresponding to the coordinates in b3 then we will write the values in this order first we will write the all 0 vector then we will write 0 0 1 then we will write 0 1 0 and lastly we will write 0 1 1 and then we will write 1 0 0 then 1 0 1 then 1 1 0 1 1 1 in what follows we will always write the table in this order this has a very useful connection to decimal numbers what we can do is that we can say that each of these vectors or strings or elements can be related to a number which is given by x1 plus 2 times x2 plus 2 square times x3 now let us evaluate each and every point when it is 0 then of course the result is 0 when it is only 1 the result is 1 but when it is 0 1 0 then it is 2 when it is 0 1 1 now let us check when x3 is 0 so that is 2 square into 0 then x2 is 1 that is 2 x1 plus x1 is 1 that is 1 so this gives me 3 this is 3 the next one is 2 square into 1 plus 2 into 0 plus 0 which is equal to 4 continuing in this way we will see that the rest of the elements correspond to the decimal numbers 5 6 and 7 so this is something that we will be calling a decimal code now of course we can extend this and go to a situation when there are n Boolean variables so bn is typically the set consisting of x1 and so on up to xn where xi belongs to b for all i belonging to 1 2 so on up to n now we will be writing the elements of bn in the order which is exactly the extension of what we have seen before so the first one will be all 0 so the decimal coding will be 0 then it will be all 0 and 1 so the decimal coding will be 1 then it will be all 0 and 1 0 so the decimal code will be 2 and then all 0 and 1 1 so the decimal code will be 3 it will go on in this way and eventually we will have the pattern which consists of only all 1 and this is in decimal code it will be 2 to the power n-1 and if we see carefully this will give us all the positive integers from 0 1 2 2 to the power n-1 we will be using this ordering always in this in these lectures and otherwise also in the theory of Boolean functions and Boolean algebra this ordering is predominantly used now what we realize coming to this point is that we can talk about something called Boolean expressions a Boolean expression is any formula that we can build up by using Boolean variables plus dot and complement that is a Boolean variables or and and complement or conjunction disjunction and complementation so for example we can suppose we fix the number of Boolean variables to be 3 so we are considering here x 1 x 2 and x 3 only we can build up any expression let us call one expression like this as g x 1 x 2 x 3 which is equal to possibly x 1 x 2 complement plus x 3 well plus x 2 now what we can do is that we can put as inputs the values of x 1 x 2 x 3 from the Boolean algebra b 3 and see what happens let us try that now so I write x 1 then x 2 and x 3 I put 0 0 0 then the next entry is 0 0 1 the next is 0 1 0 the fourth one is 0 1 1 fifth one is 1 0 0 and 1 0 1 then 1 1 0 and then 1 1 1 so I have got all the 8 entries over here and I will put the inputs in G according to that the first one is 0 0 0 0 0 so here x 1 is 0 x 2 is 0 complement so 1 plus x 3 is 0 and x 2 is 0 so it is 0 so I will write it over here so I am designating the right hand column as the function g then let us consider g of now x 1 is 1 rest are 0s now x 1 is 1 and see here x 2 is 0 so 0 complement is 1 therefore I have 1 over here plus x 3 is 0 and x 2 is 0 now this gives me 1 so I write 1 over here in the third one is 0 1 0 here x 1 is 0 then x 2 complement is also 0 then x 3 is 0 and x 2 in this case x 2 is 1 therefore again I will get 1 so I write 1 over here now we come to the fourth one that is 0 1 1 remember that we are starting from this way onwards x 1 is 0 x 2 is 1 complement so it is 0 then x 3 is 1 then x 2 is 1 so I again get 1 I write 1 over here then I have 1 0 0 so x 1 remembering that I am x 1 to x 3 is in this direction so x 1 is 1 x 2 is 0 complement so it is 1 then x 3 and x 2 both are 0 so I get 1 I write over here then see g 1 0 1 so it is x 1 is 1 x 2 complement is 1 then x 3 is 1 x 2 is 0 this also is 1 and then g 1 1 0 is x 1 is 1 x 2 complement is 0 then x 3 is 0 then x 2 is 1 this is again 1 and g 1 1 1 we get x 1 is 1 x 2 complement is 0 x 3 is 1 and x 2 is 1 so again I have got 1 so I get a pattern like the one I have got in the right hand side but this is a truth table of a Boolean function therefore what we have seen is that if we have a Boolean expression then we have a Boolean function if we have a Boolean expression consisting of n Boolean variables then it corresponds to Boolean function from bn to b and of course this function is unique so we see that a Boolean expression corresponds to a Boolean function now the question is the other way round if we have a Boolean function can we construct a Boolean expression the answer is yes and we will see an example how it works so we now consider our small Boolean algebra b2 so not bn but just b2 and we write down an arbitrary truth table involving b2 suppose x1 is again the less most variable and x2 is the next variable so the input values are 0 0 0 1 1 0 and 1 1 and let us consider the function 0 1 0 1 let us call it f our goal is to find out a Boolean expression corresponding to this function so I write here our goal is to find out a Boolean expression corresponding to f what we do here is a very standard trick we extend the table a little bit here again I have got x1 and x2 I write the input values and this is my function f so this is 0 1 0 1 I write some so called sub functions maybe yeah so these are f1 and f2 so what I have done over here is that for each one in the truth table of the function f I have constructed a function so for this one I have a function f1 for the next one I have a function f2 and if we take f1 plus f2 then we will see that we will arrive at the function f because f1 plus f2 the first row is 0 the second one is 1 then 0 and then 1 it is of course clear to us that f is equal to f1 plus f2 now the question is that can we find out an expression for f1 and f2 if we can do that then we have a an expression for f to do that we concentrate on the function f1 we see that f1 is a function which is 1 at only this input point and otherwise it is 0 the question is that how can we do that and a simple observation shows that f1 is nothing but x1 and x2 complement what I am doing over here is that I am checking the point at which f1 is 1 at that point I am checking the corresponding Boolean variables we see that corresponding to x1 the component value is 1 therefore x1 is kept as it is I see that corresponding to x0 the component value is 0 therefore I take x2 complement now if I give the input 0 1 to this function say f1 I put x1 equal to 0 I am sorry I put here x1 equal to 1 so I have got x1 equal to 1 and x2 equal to 0 this gives me 1.0 complement that is 1.1 this is equal to 1 so at least I know that this expression that I have written this expression evaluates to 1 at the point x1 equal to 1 and x2 equal to 0 now if we scan all the other points we will see that this pattern will never match therefore in other places either x1 will be 0 in that case f1 will be 0 or x2 will be 1 in that case x2 complement will be 0 for example if I evaluate the same function f1 at let us say x1 equal to 0 and x2 equal to 1 this will evaluate to 0.1 complement which is 0.0 which gives me 0 we can check that this is going to happen in all the other input points now so what is the rule of getting these functions these functions which we may call component functions of f or at this point or whatever these functions are one only in one input points and 0 in all the other input points so check that pattern and wherever the Boolean variable corresponding Boolean variable values are 1 keep the variables intact corresponding to the other entries just take the complement of those variables take the product of them and get a essentially a product term of some Boolean variables and their complements now in the same way we can get the expression for f2 which is f2 x1 x2 now please see that f2 is 1 only at the input point 1 1 and so all the Boolean variables are 1 over here so I will simply write this is x1 x2 and as we have seen if we take or or plus of these two columns we get back our function f so we can write f1 x1 x2 equal to x1 x2 complement x1 plus x1 x2 we do not write the one here because this is the whole function f so we see that f is x1 x2 complement plus x1 x2 we can evaluate this function here and check that it is indeed so so let us take the first point when both are 0's then x1 equal to 0 x2 equal to 1 and x1 0 so I get 0 then when x1 equal to 1 and x2 equal to 0 then this is 1 and this is 1 so I get 1 over here when x2 equal to 1 and x1 equal to 0 then this is evaluated to 0 and this is also evaluated to 0 so I get 0 over here and when both are 1 then this is 1 therefore I will get 1 over here so we see that the expression that we have got matches with the Boolean function this example more or less convinces us that we can get a Boolean expression for any Boolean function on any number of variables the basic rule is same just decompose the function into functions having one at only one place and correspondingly write the product terms and add them up at the end here what we see is that we are getting the Boolean expression in the form which can be expressed as sum of products indeed we sometimes call these these expressions as sum of products we also see a salient point over here that is each product term contains all the variables so we come to a definition which is called mean term a product term which contains each variable in their complemented or uncomplimented form is called a mean term similarly we define something called max terms which we will be discussing shortly so I will define them in the same go a max term is a sum term which contains each variable in their complemented uncomplimented form is called a max term what we have seen just now is that the function f x1 x2 is x1 x2 complement plus x1 x2 here the salient point of this expression is that this is to salient point of this expression is that it is sum of mean terms this function f is from b2 to b so only two variables are involved and among these two variables both the variables are involved in all the product terms whose sums we are taking so these are mean terms and we are summing the mean terms to get the expression of the function and if such is the case then this expression is called a conjunctive normal form I am sorry a disjunctive normal form so this is called a disjunctive normal form or in terms DNF I repeat again if a Boolean expression is written as sum of mean terms then it is said to be a disjunctive normal form of the corresponding Boolean function now we will soon see that we can write a function as product of some terms that is product of max terms in particular and that will be called conjunctive normal form or CNF to do that we will take up again the same function that we have discussed just before and try to find its conjunctive normal form or in other words try to write a Boolean expression which is product of max terms so we look at the truth table again and we have 0 1 0 1 0 1 0 1 now what we do is that we now try to locate the zeros so we have now a function g 1 which is 0 in the first place and 1 in all the other places and a function g 2 which is 0 in the third place and 1 in all the other places we will show that g 1 and g 2 can be written as sum of the Boolean variables which are involved in this Boolean algebra and if we take the product of g 1 and g 2 that is and of g 1 and g 2 then we will get 0 1 0 and 1 which is nothing but our original function f now the question is how to get that so our rule is as follows we have g 1 we write x 1 x 2 we want it to be 0 when the input is 0 0 in order to do that I will just write x 1 plus x 2 because this expression is 0 when both x 1 x 2 are 0 but if anything else happens and it is not 0 and it is in fact 1 similarly we concentrate on this input vector which is gives me value this so it is g 2 x 1 x 2 and this g 2 x 1 x 2 is going to be 0 if I put well as x 1 intact because anyway x 1 value is 0 over here and instead of x 2 I put x 2 complement please check that these two are the functions that we need and the rule is very clear that we have only in one input value that we have to consider check each component if the component value is 0 just write the corresponding Boolean variable in the sum if it is 1 write its complement if you do that you will get exactly the function you need and then f x 1 x 2 is g 1 x 1 x 2 dot g 2 x 1 x 2 that is equal to x 1 plus x 2 dot x 1 plus x 2 complement and this is what is called the conjunctive normal form of the function f conjunctive in saw in short CNF of the Boolean function thus we have seen two very important normal forms of Boolean functions now we will move on to another useful form which is of course very difficult to write when the function is large but when the function is not on a very large space we can write that and this relates to the decimal codes that we have discussed before we will check this way of representation by an example so we have an example on a function from v3 to be let me write down all the all the points of b3 along with the decimal coding now we write the variables as x 1 x 2 and x 3 as before so I write 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 and 1 1 1 and now I write the decimal code which is 0 1 2 3 4 5 6 and 7 and suppose my Boolean function f is something like this 0 1 0 1 1 1 0 0 what I can do is that I can simply write this function with a summation notation and put within bracket all the points at which the function is 1 so it is 1 at 1 so I write 1 over here and then it is 1 at 3 so I write 3 over here 1 at 4 I write 4 over here 1 at 5 I write 5 over here of course this representation may not be convenient if the Boolean function is on a large number of variables but for a small number of variables it is going to be convenient and a very common problem is to write the CNF or the DNF of a function which is given in this way and this is what I would like to propose as an exercise to you so in this lecture we have discussed one Boolean functions truth table then we have discussed Boolean variables and Boolean expressions we have discussed decimal coding and representation of Boolean functions and finally the most important fact of Boolean functions is DNF and CNF that is disjunctive normal forms and conjunctive normal forms and their interrelations this is for today thank you


To celebrate First Generation Student Day, we asked First Gen students for helpful pieces of advice. How important it is to integrate yourself into Marquette and the communities and also find your interests here. So one great way to do that I think is to go to OFAS and not just walk around and actually go to tables to interest you so that when you start off as a freshman you have a basis of things that you can rely on. Some helpful advice that I would give First Gen students is honestly to pray more. Being a First Gen student is a lot of new obstacles coming your way for the first time so being able to reach out to your advisors, faculty and staff would be helpful because they want to see you succeed in college. Give yourself a mentor like a professor or faculty member who you trust to help you navigate through everything.