Hi guys, what is up? Welcome to my channel. So for today's video, this is one of my favorite videos to film This is my Sephora holiday bonus event recommendations So if you want to see the best products that I think you should get for the bonus event, then just keep watching There's really something fun for me about Recommending products to you guys for these sales. I just for those of you looking to spend money The fact that I can help you It exhilarates me quick deets about the sale. Like I said, this is the Sephora holiday bonus event It is going to be starting November 1st Rouge members can use their 20% discount VIP members can use their 15% off discount And then just regular insiders you guys get 10% off Rouge and VIP it starts from November 1st to November 11th Insiders it is November 7th to the 11th for you guys use code holiday save to redeem online This is available in stores and online as you guys know the best makeup comes out over the holidays So this is the perfect time for you to pick up some of the holiday makeup that you've been wanting and also to pick up Gifts for others. So the way I made my selections for today's video are I just picked only holiday Releases as this is just a holiday bonus event So I wanted to talk about my favorite holiday releases with you guys and one side I think are just really good and worth picking up you get a little bit of a discount on top of that And I think you will genuinely love these products that I'm recommending just kind of looking at what I have sitting on my desk We have a lot of eyeshadow palettes because you guys know and that is my weakness and I mean in the holidays The best stuff that comes out are the palettes. Let's start off with my girl Pat Of course, you guys know I'm going to recommend her newest biggest Mothership palette it is quite pricey Which I can understand why a lot of people don't want to purchase these full price So I always recommend picking up any of these mothership palettes that you've been buying while you can get a little bit of a discount But of course this is her newest big mothership palette and honestly I wasn't too swooned away by these colors at first I do find myself reaching and loving for this palette because the colors are so wearable You aren't going to regret purchasing this palette is so stunning You have her special formula here and you just have generally a good wearable colors This one is amazing, and I don't want to talk about it too much I'm just gonna link my review down below for you guys and you can see an action for yourself Definitely if you have been eyeing this palette, this is the perfect opportunity to get it I love the Midnight Sun, and I definitely recommend the Midnight Sun But if there's some older ones that you've also been eyeing over that one This is the chance to get any of those and she did release her holiday Collection came out with three eyeshadow palettes. Now the one I'm recommending to you guys is nocturnal nirvana This one is the best one if you ask me I think it's the prettiest one and it's the one that really excites me now These are $60, so you definitely want to get a little bit of a discount while you can This is a way to get her super special formula in a quad and I know these colors aren't for everybody They're so pretty you guys and the way they look together on the eye Like I can't not recommend this to you guys because it's truly my favorite. It really is. It's my favorite Sorry, I just checked online. These are $65 so Get some savings on them Also, I do want to give a shout out to ritualistic rose because of the three This is the one that I'm going to be wearing and using the most so while I do love nocturnal nirvana the most I mean look at these colors. If you love a pink sparkly eye, you're going to love this I'm not wearing this right now But I have a pink sparkly eye on right now because I love those shades So that is why I am recommending this one because this one is a little bit more wearable So if you are looking for more wearable colors I recommend ritualistic rose if you like a little bit of brighter colors or I want to experiment more I would definitely recommend nocturnal nirvana and that's not to say I don't like iconic illumination It's just my least favorite of the three So if you're really planning on what to spend your money on I would recommend the other two This one is a little bit of an older palette, but I just have to talk about it briefly the Natasha Denona metropolis palette This is a good value for what it is in the Natasha Denona line anyways But if you do not have it and you want it great opportunity I just think if you want to try Natasha Denona, this is such a wonderful palette to have you get such a range of colors It's a little weird to look at at first But I've been creating the most pretty looks with this palette and I just think it's the most Perfect fall winter palette just seasonally. I love it for this season. The quality is a one on these and I don't know I just I can't express my love for this palette enough I love all Natasha Denona palettes basically, but there's something about this that just gets a little bit of extra love for me Just because of all the colors you get the value and it's the quality is just phenomenal on this So let's talk a little bit about the ABH palettes. I would definitely say don't pick up all five They've released in the last two months. I'm not going to recommend all five to you I will say they all are great, but here are my personal picks the Jackie Aina palette I mean this one did not get its real moment to shine before all the other palettes came out But the colors in here are so beautiful and I know by this time most of you have picked this up But if you haven't I just want to remind you that this palette is still out It's still new and it is a wonderful palette. It's stunning It would make a great gift if you already have it for yourself and if not, it's a great gift to yourself So that's one I would recommend getting if you have not picked it up already Now I know you guys are gonna ask about the Carly Bible palette. I'm gonna be honest. I haven't even used it yet I have it. I just think between the two. I like the Jackie Aina color story better. That's not to say this one isn't good It's just personal preference if you want to pick up both go ahead But this one is a little bit better for me personally, but Carly Bible is good If you know you like that color story then get it. ABH of course did release their three big ones Do not waste your money and get all three unless you are like a true makeup artist and need these colors sure if you've been wanting all three go right ahead, but No, you do not need all three Honestly, you probably don't even need one But I would recommend that you do pick up at least one of these three because the formula on them is So good So I would recommend you pick the color story that speaks to you the most and if none of these color stories speak to you Don't pick it up for me personally The one that I'm going to recommend to you guys is the Norvina volume 3 and I know she looks intimidating But this palette allows you to play but it also allows you to wear actual neutral looks as well When it comes to you guys, I think this is the one you are going to like the most though I'm not gonna lie volume one is low key my favorite of the three So if you are into this color story, I totally recommend this one volume two was just not for me If you ask me it was the hardest one to work with of the three But I would recommend if you've been wanting one of the big giant ABH palettes pick up at least one because the quality is Super good. So whatever story speaks to you pick that one up since you can actually get a discounted price on New makeup I always Recommend looking towards the luxury items that you've been wanting. So let's talk about my guy Tom Ford So he came out with a Soleil nage Collection and he came out with three quads I have two that I'm personally recommending to you guys this first one is my favorite of the three I think it's the one that I'll use the most and I didn't expect it to be but I love the Soleil nage Quad the look that I created with it was my favorite look of the three palettes And I find myself going for more cool toned smoky eyes anyways towards this time of year So this one just really is winter and snowy to me I feel like the looks I create with this palette are even more pretty on the eyes than they look in the pants Scoop this one up if you can because it's so good if you're going to spend that much money And you just want something a little bit more unique I would recommend Soleil moon because this one is like a really icy wintery palette But I would say it's quite unique especially compared to the other Tom Ford eyeshadow quads So this one isn't my favorite of the three, but it's the most unique so as a gift This is a very unique quad to be giving and honestly, it's just very easy on the eyes I do enjoy looking at this one So let me briefly talk about the hood of nude obsessions eyeshadow palettes now I'm not gonna lie. I've only used one the one that I've used is nude light Which is the one I use on my eyes today, and honestly, I don't really recommend this one. So just So, you know if you're looking at the three, I would not pick this one up I'll go into that in another video as to why but I have to recommend picking up either the medium or the rich Even though I haven't personally tried these just from watching other videos and seeing the swatches I think these two are going to be the best ones I don't want to talk about them too much because I haven't even tried them But I wanted to recommend them to you guys because if I hadn't purchased these already I would have purchased these in the sale because they are such stunning colors They're brand new releases and they're really beautiful. So this is just a reminder that you can pick these up for the sale Also the hood of mercury retrograde It should be available on Sephora while the VIP sale is going on I will be picking that up during the sale and I recommend you do as well if you want it Obviously, I cannot speak up the quality But I purchased basically every palette that hood up comes out with pick it up during the sale if you're going to pick it up I would recommend putting it in your basket as soon as possible Because I feel as though it'll probably sell out because everybody's going to try and get it on sale Believe it or not. We're done with the shadows. So let's talk about some more face products a lot of powder products But let's talk about the hourglass ambient lighting edit ghost palette. I mean, this is beautiful I would recommend checking out my video on this if you have not yet seen it, but I did thoroughly enjoy this palette It wasn't my favorite, but I look forward to these every single year. This isn't perfect, but I love it It's so beautiful and soft on the skin. These are $80 every year.  This is such an exciting palette They also came out with a blush quad and I don't recommend you pick up both because quite frankly You don't need both But I would recommend picking up one or the other if you feel like you just generally have a lot of hourglass powders or you really like the blush formula They do have a blush palette as well, which I thoroughly recommend I personally have this because I just love the blush formula from hourglass So I'm loving this palette. I think the colors in here are beautiful This is great for travel for me because these are the best blush formula. Don't need both, but whatever your needs are This is a good time to pick up one of the two. I've been loving them and you guys it seemed quite curious I just dropped my Tom Ford palette I do not know how people have like hardwood floors in their beauty room If I had hardwood floors all of my makeup would be broken. I'm too clumsy for that. Thank goodness I have carpet. Okay, so a lot of you guys have seemed very curious about this palette when I mentioned it in a haul This is the Laura Mercier soft cheek canvas now. I just love this face palette It's probably my favorite that's come out this year and it's because it is just so pinky toned So if you don't like like a pinky blush, you are not going to enjoy this palette But if you're like me and you love a good soft pink flush on the cheeks, this is so beautiful I love the formula of this. It's what I have on right now today I use this on my cheeks and then this as the highlight I don't know just the aesthetic of it all and how beautiful it is. I definitely recommend it I did not want this at first when I saw it, but once I saw it in stores I was solid so just keep that in mind that it is much prettier in person than it is online Because online it absolutely looked like something I didn't need but in person it was a different story So I'm loving this. I think it's awesome And if you've been eyeing it which a lot of you seem to have been just so you know we get some thumbs up It's really good and the last couple things are from Fenty Beauty You guys seemed curious about these products. I'm here to talk about them with you guys So this is the Fenty Beauty bomb baby 2. It's a mini lip and face set So I think this is an awesome deal and it's really neat because you get colors that are not in their original line So if you like the formula of the highlighter and the lip gloss from their line Definitely recommend you pick this up. They have a fabulous lip gloss formula and a fabulous highlighter formula So the fact that I could get two unique colors for a really good value. I mean Duh, so it comes with this highlighter in Sandcastle and this is a b-ming highlighter So just be aware if you have texture this really shows it off But if you're putting on highlighter already you should kind of expect that but like this one really kind of emphasizes texture but I have it just right on the tips of my cheekbone so that it really Pops because this is so freaking beaming. It's a really unique color And if you like a really bright shiny highlight, you are going to love this color You also get a really pretty light champagne Gloss I love throwing these in my purse And this is a great color to highlight your lip and also just to put over any lip liner Really so I really like this color. It's quite universal I think this is just a great set in general and it's a great value without already getting the discount you guys seem super interested in this one as well and The colors in here are fantastic. All right, so time for my last recommendation for you This is the Fenty Beauty glossy posse set this I had no problems not getting a discount on this and getting it right away Because this is the most awesome set that Fenty has this year They have top of the line lip gloss formula and now you can get it in a bunch of colors because honestly Her line didn't have these lip glosses in a lot of colors and I prefer spending less for less product Anyways, these are great to keep in your purse.  You have a chocolate gloss and iridescent gloss then you have fussy Which is from the regular line, but it's one of the most universal lip colors and this hot pink one Which isn't as hot pink and scary as it looks So I think the set is wonderful all of the formulas are the same Fenty gloss formula just with a little bit of tint I know this is crazy, but it was one of the value sets that I was the most excited for this year So if you like that lip gloss formula, definitely pick that one up So that's all I have for my holiday bonus sale recommendations There are so many great holiday products out that I definitely think you should check out But just of the ones that I purchased these are the ones that really are worth the money to me and that I genuinely Think you guys will love. Let me know what you plan on picking up for the sale Whether it's a gift or if it's for yourself I don't think I've mentioned this before but if you guys shop the links I have in my bio I actually get a small commission from that which of course it goes to my channel nothing else I'm just improving my content So if you fill your bag up using my links, then I can continue to create awesome content for you guys But of course don't feel obligated to if that just makes you feel not good inside You don't have to do it I don't care just for those of you who would like to go out of your way It is a free way to support me. So that's all we have for today's video It was so fun to film. I just holiday makeup It's so exciting to me. So thank you guys so much for watching my video If you are not subscribed to my channel yet I hope you do take the time to do so and I will see you guys in my next video. Bye guys. Have a good one


this is your new drum set look what you could do no I know that's a little overwhelming he's getting scared I didn't get this for you I got it from man hello my fellow sniffers flighters and hatchlings my name is Marlene McCohen and welcome to my channel this of course is the ever so amazing gorgeous Leo take a look at Leo today before we get into our video I wanted to tell you guys something just to clear up some things for those of you who don't know or can't put this together because I totally understand why not on for those of you that want to get right into the video we will put a time code for you to get right to the video but I just wanted to address some things videos are filmed in advance to any video that you have seen on my channel up until now has been filmed before the coronavirus COVID stay-at-home order feel like I have to say this because sometimes you guys see things happening in my house and you're like don't you guys know about coronavirus yes we do in fact funnily enough we knew about it before anyone else and started bunkering down planning and being organized and we're very very strict here about it before anyone because we have a family member who is a futurist and was able to prepare us a very long time ago so that was very helpful actually you can check out the website that he created called germ info dot org I'll put the link below and it has all the latest information so this is something we knew about a very long time ago if we do have to go out we are fully covered I never have to go anywhere because I just kind of you but if any of us have to do a grocery store run or anything like that it's masks gloves and we wipe everything down before it comes into the house now you will see other people in my house they are not visiting they are living here my brother as you guys know moved here as well as CEO this happened before the situation I also have my cousin here so any video you see going forward we are all quarantined together rarely need to go out and for those of you wondering because I feel like I should address this to you guys know that we have parrot station dot com everything that we send out from our website you can rest assured that we do everything to the max to make sure everything is super clean you guys know you guys see I have like so many birds and you guys see how clean my house is most of the time so you guys have mentioned it before we're very psycho about it no one is allowed in the merchandising room only one person that's Megan everything is in contain sealed boxes and it's been there for quite a while a long time like most of the stuff so you guys can be rest assured that we're taking every single precaution possible for the boxes feathered fun boxes mask gloves and you know just minimal amount of people and everything I thought I had to address that just so you guys know because I know that a lot of you think these videos are filmed and then put up in a date but a lot goes into it I have to film this then maybe tomorrow film some more pieces and then send it and then transfer it and then like it has to be edited then it has to be looked at then it has to be uploaded no things are done so far in advance here now but the other thing I wanted to address to let you guys know also you probably hadn't seen me be very active on Instagram story and I normally am part of the reason is because it seems like I broke my rib but I did not go to the doctor because I did not want to go next to you know any doctors or urgent care and even so they wouldn't have been able to do anything so I've been kind of out of it and just so tired for the last few weeks that's why you'll have seen some compilations up on my channel I just was in so much pain guys and I just didn't want to chance it okay anyway moving on to our video I have something really fun planned for the day I have some toys that I bought for the birds and this was before I feel like I'm gonna have to keep saying this but hopefully I don't really guys can get the point now I bought a drum set for Merlin because he loves drums and he loves drumming with George so I thought it'd be really fun today to pull that drum set out and see if I could teach him to drum I think it'll be fun and see if he plays with it and the funny thing is see oh it said no I think he's gonna make too much noise and the funny thing about that is like imagine how we are with our birds in our head so much so that she's convinced that the bird will be like a teenager in a band like no no we're gonna hear so much noise coming from the garage or something anyway here we go man do you want to see if Merlin wants to play with his drum set hello hi did you give it to me yeah I know okay let's do it it's time yeah let's go okay are you mad because you can play the drum set to you can play the drum set and are you ready to go what do you guys think you think he'll learn how to play it I'm hopeful we're gonna turn it on and we're gonna surprise him right now I think he's gonna like it guys because he's kind of like you know down with the button dance and drums with George kind of like this homeless I don't want to say anything before but like we made him be like an explorer or a treasure hunter but let's be honest he's got a tent he's got a shopping cart he's got coins in the shopping cart he loves cams and the garbage and he plays drums on pots and pans so we've upgraded him today we're gonna bring Merlin out and if he doesn't want to play with me you're gonna have to play with him Merlin's gonna be thrilled I'm telling you do you really know that because I feel like the metallic sounds more you mean like yeah oh my god it's probably so true it's gonna be okay I'm sorry I'm sure he's gonna love it he better this is like you know I think somebody's gonna love it instructions in case don't need those gift for you leapfrog Academy I mean if he gets smart we'll do that all right this is it do I bring him here what is that piece oh oh what is it it a stand yeah it makes it so it can stand on like you're kidding me and now do you think he'll be using these or just this is your new drum set I know it's a little colorful for what you're used to but look what you could do no I know that's a little overwhelming but look check it out we got it no let's do freestyle no okay no we want to freestyle he's getting scared I didn't get this for you I got it for a man do it together you can have a snack but I expect you to know a full song by dinner technically we should have a cooking show because we make the best food here now after your snack you've got to learn the drums this flock mentality you'll see like when one stop you know the alphabet that's nice they both kind of start doing the same thing at the same time


What are you doing?  We are on to episode six We are more than halfway through of means of her season dose Fantastic so far if you're not liking it. I don't I Question your taste and what you actually I agree Probably the same because it's true True true gangster films. Mm-hmm focus on the people in the story. Yeah, there's there's some nice kills Obviously and and stuff like that, but more it's about the family and the story Behind it. It's not all just about the chopped off horse heads right You get those are great to have That's not central. No But yeah, we're really enjoying this but episode six. It's called and I guess the last one was Called lame for lame Tiagi probably right Yeah, I think a combination of both Tiagi and how Guru is walking around with a cane, but he's actually kicking some butt. Yeah, this one's called on uncouche You can come back up here come on which I believe it says if the internet is correct. It means curb Curb like are they gonna curb check them? Ouch, are they gonna curb his enthusiasm? If you don't know what curb checking is it's not fun. It's not a fun thing But Yes, anyways, as always Amazon lets us use five minutes of their episodes per reaction Cut up only five minutes. That's Amazon's rules. I cannot change that I just say that every video in case people are new here and don't Understand it's a copyright thing Next if you'd like to see the full uncut version head over to patreon. It's all the episodes are there There's future episodes there as well. So go check those out Rick. Are you ready? Yeah, Lance is such a good boy Me and such a good dog Yes, I'm ready. Three two one I'm looking for a guru. Oh Look at her They're gonna make it explode They're gonna blow impossible Nice I Like how in movies and television shows Like you always see people just walking slowly away from an explosion Yeah, that would never happen never happen.  What do I think he's gonna be killed very shortly? I don't think so.  In other words, let's do a pine his back Yeah, younger brother seems to be neglected He'll do a behind his back Everybody is so uniformly good in the show Mm-hmm So he's not only a dawn but now also a politician Wait I know So good. Yeah, everybody in this show is really really good So the title's name was if this is correct curb Like maybe cuz he curbed his brother. I Don't know. I was trying to well fit and and the end there trip at these curbing Guru in terms of you know Guru was just looking at those poppy fields Trip at these got him. Yeah no trip and Guru's thinking we're gonna be able to have our own opium business and Trip at these working at the top and undercutting him So I think there's a lot of curbing going on here like people who think they're gonna get in They're gonna get something or getting shut off by somebody else somewhere, you know Yeah, it's just so well-written. I think Moona also messed up. I'm obviously a lot but Now she's definitely going to be helping trip at these wife and Oh, yeah, and taking down the family like he didn't like she was actually like I feel like she loved him Like Moona and she was like really excited and then she was like, all right.  I'm really interested now that Vijay Varma is playing twins twins I've always I've always wondered because you know, there's been Throughout cinema history multiple people have played multiple people in in a film Yeah, it's it's always such an interesting thing like because You obviously as an actor you have to get into a mindset and when you're on set Now you have to get into two different mindsets on the same set Yeah, and then when you're doing work where you're having to interact You're not getting to feed off of each other, you know, that's why for me the best twin performance I've ever seen is Tom Hardy. I just I'm you forget you're watching the same actor in both of those roles. He's he's he's that freaking genius So obviously Vijay hasn't had anything in the script to give him anything like that as of yet But no, it's a cool. It's good. It's cool that he got to play the twin brothers You know the way that Paul Dano did and there will be blood. It's similar to that right now Yeah, and I like I like his dad as well, whatever whatever that actors name is. I feel like he's a he's a good actor I don't think we've ever seen him before No, and you know what I was thinking too Ali told us he didn't give away any spoilers But he did say Guru takes a different direction because this is a very different He's still got some of those traits But he's a much that wedding shooting the loss of his brother and that tragedy he's much more methodical It's not all just I'm gonna go kill people more mature.  Yeah, it's just such a well-written show. Yeah Man, we literally I've literally gotten to the point, you know in the first season. We were like, okay Where's the next kill? I Don't even care anymore. I just am so interested in the characters in the story that a kill is just gravy Yeah, and they're the past two episodes have been Some of the in terms of action some of the slower ones in the in the whole series and I'm sure Obviously they added one episode from from last season So obviously they they have more time to expand on Everybody's relationship and all that right, but yeah, I have not felt bored one little bit. It's it's a it's a great show great acting and great writing and I'm loving it. Yeah, so that was great. We only have four episodes left. That's the sad part From this season, maybe we'll have to be like everybody else and wait a couple years as opposed to just a month for the We we got a little spoiled With it, but if you'd like to see the full uncut version of our reactions, you can head over to patreon And there's full on over on our long reactions of us Just staring at the screen If you'd like to see it invigorating you get to hear Rick eat whatever he's eating But anyways


Continuing our discussion of transpositions, earlier we showed that if I can write the identity as a product of transpositions ending in UV, what I can do is I can sweep this element up to the first transposition, and because this is the only time this element U is moved, this product can't be the identity. And the problem we ran into is while that worked if we had an odd number of transpositions, it also worked if we had an even number of transpositions, and it suggests we can't actually write the identity as a product of transpositions. So we know that the identity can be written as a product of an even number of transpositions, so something keeps this from happening. Somehow the transposition involving U vanishes, and it seems that we need the following lemma. Let E be expressed as a product of K transpositions ending in something. The process of moving the transposition involving U to the front will invariably eliminate a transposition involving U. That is the only way we can prevent this situation where U gets all the way to the front, because if U gets all the way to the front, we cannot write the identity as a product of transpositions. So somewhere along the way a transposition involving U has to be eliminated so that you can't get to the front. Well actually the proof of this is pretty easy. If it doesn't happen, we can't write E as a product of transpositions. So if E can be expressed as a product of transpositions, then somehow U has to be eliminated because otherwise we can't do it. Now we should actually look a little bit beyond the proof. The proof is a very very very very simple proof, and we might actually ask ourselves, how does it happen? We know it has to happen, but how can a transposition involving U be eliminated? How do you eliminate such a transposition? Well we can take a note from our earlier observation. Back when we assumed that the identity could be written as a product of K plus 1 transpositions, we considered the last two transpositions, but we did assume that they were different because if they are the same, their product is going to be the identity, and we don't have to include them. And so that means that the only way we can really eliminate a transposition of the form UX is somewhere along the way. We have to encounter two identical transpositions, UX, and the one before it is also UX. And if we do that, if we do encounter those, their product is the identity, and we can eliminate both of them at the same time. And that means that this product of K transpositions, in order for it to be able to express the identity, this product has to include somewhere along the way this product of a transposition with itself, and that means I can eliminate two transpositions. And so while we did take the easy proof, if we look a little bit deeper, if we go farther into the proof, a better proof gives us a more useful result, not only do we eliminate a transposition involving U, but in fact we eliminate two transpositions. And so we reduce the number of transpositions down to K minus 2. And so let's go back to our original claim. Suppose we could write E as a product of K plus 1 transpositions. Well, it follows that we should be able to eliminate a transposition involving an element. We should be able to eliminate two transpositions, which knocks us down to E as a product of K minus 1 transpositions. But, lather, rinse, repeat. And what I can do is I can take that last transposition, U, V, and I can move U to the front. I can start to move it to the front, but somewhere along the way I'll eliminate two transpositions. And now I can write E as a product of K minus 3 transpositions. And I'll repeat the process. And remember, our assumption was that K plus 1 was actually an odd number, which means that if I continuously eliminate two transpositions, eventually I'll be able to reduce this down to the identity as a product of 1 transposition, which is impossible because a transposition is not going to give us the identity. And there's our proof. Now, it's worth noting the following. We began by thinking, hey, this might be a good proof by induction because we thought, well, it seems to be that the key step here is if we could write the identity as a product of K transpositions. If we know something about K, we know something about K plus 1, or we want to prove something about K plus 1. So this looks like it's a good proof by induction. And so, because we wanted to prove something was impossible for K plus 1, we tried out a proof by contradiction. We assumed the identity could be written as a product of K plus 1 transpositions. Except we found a gaping hole in our proof. Our process implies that we could not even write the identity as a product of any number of transpositions. Now, we dug ourselves out of the hole. We figured out that somewhere in this reduction process, we'd actually lose two transpositions. Now, here's the important thing. We could actually go back and clean up our proof. And it's a good exercise to do that because if we go back and clean up our proof, the really important thing that we found is that if I can write a product of K transpositions that gives us the identity, I can rearrange that to be a product of K minus 2 transpositions that will also give us the identity. And, lather, rinse, repeat. We write this as a product of K minus 4 transpositions that give us the identity, and so on. And that means that K has to be an even number because I'll keep knocking down the number of transpositions by 2 until I end up with zero transpositions that give us the identity, or if K is odd, one transposition which is impossible. And so, if I think about that, this is the only important thing I have to prove, and then all of the scaffolding is actually unnecessary, and I can just get rid of it. But let's keep it in because here's an important moral here. The proof is more about the process than the result. I didn't need all of these steps here, but I had to go through these steps to identify what was actually important. And while I might omit these steps in the write-up of the proof, it's important that we go through these steps because they give us insight into how we get to this last important step. And so, let's tie that together. We've proven that the identity permutation has even parity. And so, here's another useful habit to get into as a mathematician. You want to look for the corollaries. What's a corollary? Well, roughly speaking, a corollary is a result that you can prove from a theorem, lemma, proposition, or whatever. But the proof is one or two lines. It's a really, really short proof. So, for example, here's a corollary. Once you know the identity permutation has even parity, then you know that any permutation and its inverse have to have the same parity. So, if I can write this as a product of some number of transpositions, if I could write this as a product of some number of transpositions, then the product of these two has to be the identity which has to have even parity. And these are either both odd or both even. Also, another useful result here is that if I have a permutation with some order, if that order is odd, then I know that the permutation itself must have even parity. And again, that's because sigma to the k has to give us the identity, and we know that's an even parity permutation. How about the actual permutation theorem? So, let's go ahead and take a look at that. So, as far as I have a permutation and I could write it as a product of transpositions, and maybe I could write it as a different product of transpositions. Well, since the permutations are equal, then the product of transpositions here is the same as the product of transpositions there. Now, I do know, because we're dealing with a group, that sigma has an inverse, so its inverse can be written as a product of permutations, and so it's the inverse. I'll apply that product of permutations to both sides. I'll multiply by the inverse here, multiply by the same thing over there. Over on the left-hand side, this has to be the identity, and over on the right-hand side, what I have is this product of permutation. So, let's think about that here. Now, I know that M and M' have to have the same parity. Their product is the identity. There has to be an even number of transpositions over on the left-hand side. But, again, because this product is also the identity, there have to be an even number of transpositions over here on the right-hand side. So, that says that M' and N have to have the same parity. So, M and M' have the same parity, M' and N have the same parity, so that M and N have to have the same parity. And that proves our parity theorem. The parity of a permutation is the same regardless of the transpositions used to produce it. If a permutation could be expressed using an odd number of transpositions, then any expression will use an odd number. If it can be expressed using an even number, then any expression will use an even number.


The students in Bagley are lending a helping hand as part of a food packaging event for the Feed My Starving Children Organization. Haiti Clotter shows us how students prepared meals one box at a time. More than 11 million people are undernourished in developing countries according to World Hunger Education Services. High school students from Trek North and Bemidji High School along with other volunteers decided to take action with the food packaging event. Feed My Starving Children is a non-profit organization that sends nutritious meals to malnourished children in more than 70 countries. Within those countries we have right around 200 partnerships. Those are people who are on the ground and they know the kids by name. They know their stories. They're able to get them these meals that are packed. The Bagley Arena turned into a food hub where more than 100 volunteers were at each station preparing meals. Some were measuring ingredients, checking the weight and then sealing the bags. About 20 or so vitamins, veggies, dehydrated potatoes and carrots, soy and then rice. Donations make all of this possible and provide the food. The goal is to feed the body of others. We don't want anybody to be hungry, to anybody to go through starvation and so mobile packs and going to our permanent site are honestly something that's really important and crucial and if we can get more people on board I mean who doesn't want to help somebody. The average session lasts about two hours. One bag equals to six meals, one box is 36 bags and one of these pallets is 36 boxes which equals to over 7,000 meals. Event coordinator Michael Davis says doing something like this unites everyone. It maybe wakes people up to the fact that hey, we've got it way better than most people so we need to get with it and help people up. At the end of the shift, more than 24,000 meals were packaged and ready to go. That's enough to feed 66 children for one year. In Bagley, Haiti Clotter, Lakeland News. And once there's enough food to fill up a container, it'll be delivered to its destination within two to four weeks. For more information about Feed My Starving Children, visit our website lptv.org. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.