QuickBooks Desktop 2023, Vertical Analysis, Profit and Laws, P&L, Income Statement. Let's do it within 2-its, QuickBooks Desktop 2023. Support Accounting Instruction by clicking the link below giving you a free month membership to all of the content on our website broken out by category, further broken out by course. Each course then organized in a logical, reasonable fashion making it much more easy to find what you need than can be done on a YouTube page. We also include added resources such as Excel practice problems, PDF files and more like QuickBooks backup files when applicable. So once again click the link below for a free month membership to our website and all the content on it. Here we are in QuickBooks Desktop Sample Rock Castle construction practice file provided by QuickBooks going through the setup process we do every time. Maximizing the home page to the gray area in the view drop down we got the hide icon bar open windows checked off open windows open on the left hand side reports drop down we're going to be opening up the profit and loss the P and the L the income statement ranging to the change in we're going to go from 010124 to 123124 customize that report fonts and numbers change the font to 12 okay yes and okay let's open up the balance sheet to so going to the reports drop down company and financial the big B balance sheet down below hitting the drop down we want to bring this then to the fiscal year of 123124 customize in the report to the fonts and numbers so we can change it to 12 okay yes okay there's the setup process we do every time we're focusing in on the P and L profit and loss income statement type of reports last time we did a comparative report doing kind of a horizontal type of analysis you might call it this time we want to do a vertical type of analysis remembering that in prior sections we did this on in essence the balance sheet and we saw that we get that vertical kind of component or vertical analysis that kind of compares everything to the total assets let's do it real quick on the balance sheet just so we can remember what happened if I go to the customized reports on the balance sheet and we hit this percent of columns item I'm going to hit it I didn't really hit it that hard I just I clicked it really I didn't really hit it but I'm going to say okay if I hit it it wouldn't it's not be alive anymore if I hit it that's what I'm talking in case so we got the total assets down below and he gave us a percent of each asset as compared to the total asset each liability and equity as compared to the total liability and equity now it's a little bit different on the income statement or the profit and loss because the total or the bottom line number in essence of the income statement represents net income assets minus liabilities so it doesn't really make sense typically to compare things to net income what we really want to be doing is comparing things to the goal of the business particularly the profit and loss that's income generation in other words you can think of the income statement as in essence income and expenses the goal of the business is to generate income revenue generation all the expenses that we are incurring are done so in order to generate the end goal of income so typically when we do this kind of vertical ratio analysis we want to compare everything not to net income but to total income so that'll be the general idea you can see the difference here when I go to the customize up top to do this to click this off when I hit it when I hit this thing we're going to go down here and you've got some different items you got the percent of row you got the percent of column I'm not going to use the percent of column this time typically what is going to be used is the percent of income meaning I want to take in essence every number on the income statement and compare it to the main goal of income statement of the business generating revenue the income line item so we'll check this one off and then we'll say okay so there we have it now you can tell that that one is the one that we're comparing everything to because it's the only one that has 100% next to it as the percent meaning you're taking this divided by itself one or 100% everything else is being compared to it so on the income side of things that kind of makes sense or similar to what we saw on the balance sheet so we might say okay well yeah if i'm taking like this construction income and comparing it to total income 447 537.34 divided by total income which is this 545 657.14 gives us about 82 if I move the decimal places over hold on a second that didn't come out right did it let's do it again 447 537.34 divided by this number 454657.14 gives us about that looks right 94.4 percent so and that kind of makes sense because we're comparing it to the category total down here and we can think about each of our income line items and think about which one of them is most important to us as compared to our total income line item that could be useful so that we can obviously identify what our big moneymaker is but also we might be benchmarking to other companies and trying to trying to mimic them typically larger companies because we're looking up to people and trying to imitate people that we admire that are making more money than us basically so so we can't really compare our dollar amounts to them but we might be able to compare our percentages to them now given that you might say well then on the income statement side of things shouldn't we be comparing all the expenses in essence like to total expenses down here you could do that but but really you on another way of thinking of it is all the expenses are there to generate the income so since that's kind of like the goal of the business it's common for us to compare everything to that income line item so we still got the income line item we can compare the total cost of goods sold which is here 180124.19 divided by the income which is 454657.14 so moving the decimal two places over we're at the 30 we're at the 39.6 about now the cost of goods sold notice it's a little bit different for a construction company but those are the things that are kind of part of the inventory if we just bought things and then we marked them up and sold them and that's how we generated revenue then the cost of goods sold will typically be a huge component of our of our income right because because whatever the markup will be representing the relationship between the cost to get sold in the income in other words if all of our income was for things that we bought marked up and sold whatever that markup would be would be reflected in the relationship between the cost of goods sold and the income that's why this sub calculation of cost of goods sold breaking it out in its own expense category is a useful tool and then of course we have all the other expenses that we can break out down below we can compare and kind of get an idea of the auto versus insurance on a percentage basis once again being able to compare these to other companies as well to see if they have similar relationships on theirs especially you can see the gross wages clearly that one oftentimes this is going to be the payroll will be a significant impact on or a significant part of our expenses and a big item in relation to the income and so we can go through there and that's going to be the general the general concept of it we're just going to be comparing everything to the income line item all right so if we wanted to clean this up then we could format it i'm going to change this to a maybe an income statement just so we can make that change which could be useful if you're providing it to somebody else we'll call it a vertical analysis income statement or something like that and then this is still good here i'll remove these items on the left we'll make negative numbers bracketed and remove the pennies let's customize to do that to do that we're going to go to the header and footer and i'm going to say let's call this a vertical analysis income income statement statement and then i'm going to get rid of the date the time the report basis let's put our name in the footer put our name in the footer and then fonts and numbers we changed it to 12 on the size let's put some parentheses around the negatives let's remove the pennies let's make the negatives read that's what our custom has been so we're going to say okay so there we have it vertical analysis another report that can be quite useful and we can batch that together as we're putting together our month end possibly our quarter end our year end reports that we might be providing to our our clients note that we could also create this column as well when we do comparative reports as we saw in the past so you might say hey i like i like that column it might as well add that in as i do like comparative reports possibly changing the date here from let's say let's say we took it from 10 oh one two four and then i did this kind of comparative report by saying totals and bring it down to a month for example it gets quite long because now you have these extra columns but now you've got november uh you you've got november you've got october november and december and then you've got the percentages as well so that's a little long so maybe like two periods would be good for that kind of report so maybe you did the last two quarters for example maybe i went from oh seven oh one two four to december the last half of the year and then said let's make it on a quarterly basis and okay so then now we've got two quarters and the percentage for both and the percent for the total maybe that's not too much and it could be digestible and give that more information it doesn't work so well if you do the other kind of comparison with the previous period because that's typically designed to use kind of like the percent change and the dollar change here but you can experiment with using those types of reports and trying to include like a vertical analysis column as well


Welcome back to our lecture series, Math 3120, Transition to Advanced Mathematics for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Miseldine. In this first video for lecture 24, we're going to talk about some important properties of relations. Now remember, we introduced the set theoretic definition of relations in lecture 23. Just as a quick summary here, we say that a symbol R is a relation on two sets A and B. If R is in fact a subset of their Cartesian product. We like to think of it as the element A relates to the element B, if and only if their ordered pair belongs to the relation itself. There's a lot of benefits from the set theoretic definition of relations, some of which we're going to see in this video as well. Now before we get into these important properties of relations, let's mention a little bit, just make sure we understand exactly what is what we're talking about here. When we have a relation, we might say something like generically, we'd say like X is related to Y or if you want more specific examples, you might have things like five is less than seven or three is equal to seven, something like that. Now be aware that these relations are in fact statements. They are either true or they're false. That's the only value that you can give to a relation here. These are relational expressions I should say. The relation itself is a set, but with a relation, you make these expressions like the following five is less than seven. This is a true statement. Three is equal to seven. This is a false statement. Then the original one X is related to Y. Currently, this is an open statement because we don't know what X, Y, or the relation are or themselves. But every relational expression is a statement. It's either true or false. Now on the other hand, if you were to take like an algebraic expression as opposed to these relational expressions, you get things like three plus four, you get two times nine, you get one divided by two. In each of these situations over here, these aren't statements, these are quantities. They're numbers and therefore they could be a real number, an integer, a rational number, complex number, depends on the number set that you have there. Now, the way that we write an algebraic expression is very similar to how we do a relational expression. You're gonna have some number, you're gonna have some symbol, and then you're gonna have some other number, right? You put those together. Now with these algebraic expressions, your symbol is this operation. You're combining the numbers together, this operational symbol over here. And so when you combine two numbers with an operation, you get back another number. On the other hand, over here, when you have your relational expressions, you connect together two quantities, which honestly any object in mathematics, you could call it a quantity, right? I mean, what's a number after all? A number is just toys that mathematicians like to play with, right?  With relations, you connect two quantities together via the relation like these ones over here, and then you don't necessarily produce the quantity back, you actually do this true-false over here. So with operations, we compute things, but with relations, we prove things, which admittedly since the value you attach to a statement is a true or false, it's a Boolean value, you can make the argument that proofs are just calculations for statements, much like algebra does calculations with operations, but I somewhat digress. It's important to realize that relational expressions are statements, they're either true or false, and therefore we often have to prove things related to relations in this situation. So in this video, what I want to do is describe some very important properties that relations can have and then start to discuss how one can prove various properties here. Because while we can look at individual statements like two are five, it would be better to think of things generically, like if I take a generic element X and a generic element Y, what can I say about the relationship, the relation as a whole and not necessarily individual or pairs? And so there are some very important properties of relations that I'm gonna list in this video that these are considered all over the place. So these are the five top properties that we're interested in when you discuss a relation. So in all of these examples, we are gonna consider relations from a set back onto itself. So R is a relation on the set X here. We say that R is reflexive if for all elements X inside the set, it holds that X is related to X. So it's reflexive if every element is related to itself. Now it turns out the antonym of a reflexive relation is an irreflexive relation. We say that R is irreflexive if for every element X of the set, it holds that X is not related to X. So what are some examples of reflexive and irreflexive relations here? For the reflexive relations, we have things like equality. This is like the poster child of a reflexive operation relation, excuse me, because every element is equal to itself. We do also have that less than or equal to is a reflexive operation relation, excuse me, greater than or equal to, but admittedly that's because they contain the equality relation inside of them, of course. You can take set containment, set containment. So a set is a subset of itself. So equality does happen, it's reflexive. You have divisibility in that situation. A number does divide itself. You can also take the approximate symbol, if you're like pi is approximately pi. That is true, right? That's a really good approximation, but generally you don't have equality, but it does allow for it. And in fact, in general, a relation will be reflexive if and only if you actually have the equality relation as a subset of that. So if the relation contains all of the equality, which of course what I mean in that situation is you have the relation X comma X for all X inside of X. If this is a subset of the relation that makes it reflexive, okay? Now in the previous video we had about relations, we did introduce this idea of a relational digraph. It's an illustration of the relation itself. And this is really great when you have like a finite set. We've looked at some examples of relations on sets with only five elements. We just called them one, two, three, four, five. And how do you detect reflexivity on the diagram, on the digraph there? It comes down to if every vertex has a loop, then the relation has to be reflexive. So for example, you look at this one right here. This is an example we considered beforehand. Notice how every element has a loop.  So like for example, there's an arrow between one and five because one is related to five and also five is related to one. And so the arrow pointing from one back to itself suggests that one is related to one. And this arrow here says that two is related to two, et cetera, et cetera. When you look at the digraph, a reflexive relation will have a loop at every single vertex. And therefore you see this one right here. This is not a reflexive relation because there is not a loop at every single vertex. This actually leads to the second topic we were talking about right now. What does it mean to be irreflexive? Irreflexive means that equality never happens, basically. In other words, an element is never related to itself. Now we've seen examples of this. Of course, you have not equal to is definitely an irreflexive relation. It's like the poster child of irreflective or irreflective relations there.  These are also irreflexive because equality is not allowed in that situation. If you take the negation of the set symbol, it turns out that is also irreflexive. Now, so this one means you're not a subset of and since a set is a subset of itself, the negation would be irreflexive. But some people also introduce this symbol right here where you take the subset symbol but you put a line through the equal part of it saying that you have to get a proper subset that would also be irreflexive. We could also mention that if you take not to visibility, not approximately, these are all also irreflexive things here. It turns out that if you take any relation, you could always introduce its negation by putting a slash through it. And it turns out that if you have a reflexive relation, its negation will always be irreflexive. So honestly, most of these symbols here are just the negations of the things we had above. Like you have not equal, you have not subset, not divide, not approximate. When it comes to our inequalities here, they actually got switched around. If you take the negation of less than or equal to, then you're gonna basically get the opposite here. So those guys get switched around. So if you ever negate a reflexive relation, you get an irreflexive relation. That basically gives you each and every one of them. So what about our symbol of element containment inside of a set, right? Can it be a reflexive operation, a relation, I should say? So if you take something like A is inside of A, this is actually forbidden in the axioms of set theory because we want to avoid Russell's paradox. So this never happens, which means this does happen. The not element, or not in symbol is actually a reflexive relation, which then means its negation is irreflexive. So it's important to include that in that list right there. With regard to the relational digraphs, notice that irreflexivity is like the opposite of reflexivity. So if having a loop at every single point makes you reflexive, then the absence of a loop at every single point makes you irreflexive. So this example right here of the pentagon is an irreflexive relation. You also have, for example, this one right here. This is an example of an irreflexive relation because there's no loops on any vertex whatsoever. This one right here, I couldn't quite fit it above. This is an example of a reflexive relation because we can see the loops at every single spot on the list there. I want to point your attention to this one here as well. So this one is neither reflexive nor is it irreflexive. It's not reflexive because we don't have loops at every single vertex. So it's not reflexive, but it's also not irreflexive because it does actually allow for loops. So notice here, it happens that two is not related to two in this example, so it's not reflexive, but we also have that three is related to three. So this reflexivity versus irreflexivity, it's not like an either or. If you're one, you can't be the other. If you're reflexive, you can't be irreflexive. And if you're irreflexive, then you can't be reflexive. Those are going to be opposites of each other. I should mention there is one example of a relation which is both reflexive and irreflexive. And this is basically the only one there is. You take X yourself itself to be the empty set, which means the only relation on the empty set is the empty set itself. This is an example of a reflexive and irreflexive relation. It's the only such one because to be reflexive, you have to take every element of inside of X and it has to be related to itself. But since X doesn't have any elements, it's the empty set, this condition is vacuously true for the empty relation. But for irreflexivity, it works the other way around too, that for all elements, you can't be related to each other. So if there's nothing, then it's not related to each other. So you're good, right? So this is an example of something being vacuously true in that situation because to be reflexive, let's say you're trying to show like, oh, is it reflexive or not? Well, if it weren't reflexive, that means there has to be an element not related to itself. No such element exists because it's the empty set, so it's reflexive. For irreflexivity, in that situation, to be not irreflexive, there have to be an element for which it's related to itself, no such element exists. So this is an irreflexive relation. So this is the only one that there's possible here. Now, when we were discussing the reflexive property and the irreflexive property previously, I introduced the notion of a negation of a relation. We did that, of course, in the previous lecture, but we made it a little bit more formal. I put the definition on the screen there, although I didn't really draw attention to it because we've already really talked about it. Given any relation, there is its negation. So the relation there has the condition that A is related to B if and only if A, B is inside the set. Well, what does it mean for A to be not related to B? That actually would mean that A, B is not in R. In other words, the negation of a relation is just, of course, the complement of R when you view it as a subset of A cross B. So given any relation, we can define a new relation, its negation. Another important relation that we can construct from a given relation is its inverse. So if R is a relation from A to B, then the inverse of A, which is denoted as R to the negative one superscript, this is going to be a relation from B to A. So notice the direction switches around. That is, R inverse is a subset of B cross A, and this is the relation such that you have the element B, A exactly when A, B is an element of R. So in other words, we have that B is related by the inverse to A if and only if A is related to B. That's the relationship there. This happens if and only if, of course, A, B is inside of R. So we have this inverse relation as well. And then, of course, you can take the negation of the inverse. You can combine those together if you want to. We weren't going to talk about that one so much, but the inverse is an important relation. In particular, when A equals B, all four of these things are relations on the exact same set. So R is a relation on A, the negation is a relation on A, the inverse is a relation on A, and the negation of the inverse is a relation on A. We mentioned when we talked about reflexivity that there's a relationship between the properties between a reflective relation and its negation, which necessarily has to be irreflective. So one property of one can influence the property of another. And this is why we want to introduce this inverse property because for the next one, the next property we're going to talk about, which is known as symmetry, turns out there's a relationship between the inverse of a relation and the relation itself when symmetry gets involved there. So what is symmetry here? So we say that R, the relation R is symmetric if for all points X and Y inside the set X, if X is related to Y, then Y is related to X. The opposite of symmetry here is what we call anti-symmetry, which is a little bit different than the irreflexivity we saw before. Anti-symmetry actually means that if you have elements X and Y, and if it holds that X is related to Y and Y is related to X, then it must have been the case that X equals Y. So basically for symmetry, if you have a relation in one direction, you have a relation in the other direction. For anti-symmetry, that never happens with the exception of equality. So some examples of symmetric relations that we have introduced previously. So equality is a symmetric relation. So if three equals two, then two equals three, which of course I know that's not true, but if you had like X equals three, X equals X, sure that could be true. That kind of led me to my next one, non-equality. In this case, if you're not equal, that is a symmetric relation. I'll say that one, two is not equal to three, that implies three is not equal to two. It's a symmetric relationship, approximately equal to. Is symmetric not approximately equal to? Turns out there's a connection between symmetry of a relation with the symmetry of its negation. Notice these are negations of each other, both symmetric. We'll see in just a second that for every symmetric relation, it's inverse, excuse me, it's negation. It's also symmetric. Okay. Now when it comes to a relational digraph, how do you see symmetry? Symmetry is parent when it's present when you can see arrows going in both directions. So every arrow is reversible. So when you look at this example here, the arrows only go in one direction. You can't turn around. So this is not a symmetric relation. You look at this next one right here, same thing. The arrows only go in one direction. It's not symmetric. But if we look at this third option here, this one, you'll notice that every time there's an arrow, it's a double arrow. So this is an example of a symmetric relationship here. We'll look at some more examples in just a moment. Let's then talk about anti-symmetry, then sort of the opposite notion of symmetry. So you're anti-symmetric that if whenever you have a relationship going in both directions, so whenever you have a double arrow, it turns out it actually was a loop. That is only equality. It's the only way you can be symmetric in that situation. So examples of relations that we've seen before that are anti-symmetric, less than is an anti-symmetric relationship here. Greater than, less than or equal to, greater than or equal to. Now I should mention that these are all anti-symmetric. Anti-symmetry doesn't say that you have to be reflexive. Anti-symmetry says that if you have symmetry, it only happens when you're equality. Equality is not required. So equality is allowed here, but equality is forbidden here, but all together these are anti-symmetric because for this one and for this one, there's never a case where you have A is less than B and B is less than A. That never happens. Less than or equal to is possible, but never for less than or greater thans. Okay, so those are anti-symmetric relationships. Of course, there is the set containment symbol. That's going to be anti-symmetric. And so if you look at the examples we've seen so far, I kind of already suggested this, that for an anti-symmetry digraph, it'll be anti-symmetric if there's never bi-directional arrows. The arrows never go in two directions. Loops, that's okay. A loop is okay. You don't have to have loops though. So when we look at these ones again, so this example is an example of a anti-symmetric digraph here. The relation is anti-symmetrical because the arrows never go in two directions. The only potential situation is a loop. The loops don't count against anti-symmetry. This one is also anti-symmetric because the arrows never go in two directions. This one, like we already mentioned, is in fact not anti-symmetric because the arrows go in two directions. This one is an anti-symmetric diagram because again, the arrows never go in two directions. And then lastly, this one right here, this is an example of one that is neither symmetric nor anti-symmetric. It's not symmetric because there do exist arrows that go in only one direction, such as one is related to two, but two is not related to one. But it's also not an anti-symmetric one because you do have by directions here. You do have that two is related to five and five is related to two. So not an anti-symmetric diagram, not a symmetric one. So you can have relations which are symmetric. You can have relations which are anti-symmetric. You can have relations that are neither symmetric nor anti-symmetric. You can have relations that are both symmetric and anti-symmetric. So we consider that for reflexive and irreflexive relations there for which I said only the empty relation can do that. Well, if you take the equality relation, so x, the only things related to each other are x and x is related to x, that's it. So if you take the equality relation, which again, this is a subset of x cross x. If you take any relation that's a subset of that, this is going to be both symmetric and it's going to be anti-symmetric. And it turns out you could show that the only symmetric anti-symmetric relations are going to be exactly those which are subsets of equality. Now I alluded to this fact earlier and I wanted to make it explicit now. With regard to a relation, if it's symmetric and then it says we can actually say something about its negation. If a relation is symmetric, then its negation is also symmetric. This is what I meant earlier, that the properties of one relation can influence the properties of its relatives. So for example, the negation is also going to be symmetric. How does one prove symmetry? Symmetry is itself a conditional statement which is universally quantified. So for all elements x and y, if x is related to y, then y is also related to x. So we want to prove that universal conditional statement there. So what we're going to do is we're going to take two generic elements, x and y, and suppose that the relation satisfied. Now I'm not talking about the relation r, I'm talking about the relation not r. So let's suppose that x is not related to y. Now I need to then prove that this implies that y is not related to x. I'm going to prove this by contradiction. So I'm going to take the conclusion I want and I'm going to negate it. So suppose that y is not related to x is that y is related to x. So suppose to the contrary that y is not related to x. Now if y is related to x, sorry, suppose to the contrary that y is not not related to x. So y is related to x. Well, this is now involving the original relation r, which is a symmetric relation. So if y is related to x, that's what we're assuming, that means that x is related to y. But that then contradicts our original one there. We get a contradiction. That's the opposite of what we assumed. So therefore, this is now false. We get that y is not related to x, which is then just this one turned around. So that actually tells us that the negation is a symmetric relation. So if a relation is symmetric, it's negation is also symmetric. This is why since equality is symmetric, not equals is also symmetric. Approximately we said was symmetric, which means not approximately is also symmetric. Now we introduced this inverse. How are they related? Well, it turns out that a relation is symmetric if and only if it's equal to its inverse. So the inverse reverses the order of things. Well, if you reverse all of the order pairs and you back the exact same thing, that's exactly what it means for a relation to be symmetric. So far we've introduced reflexive and then it's opposite irreflexive. We've introduced symmetric and it's opposite anti-symmetric. There's one last property of relations we wanna introduce in this video and this is what we call a transitive relation. We don't have a opposite notion of transitivity. At least we're not gonna introduce one in this video here. Again, if r is a relation on x, we say that r is transitive if for all elements x, y, and z inside of the set x, whenever x is related to y and y is related to z, it holds that x is related to z. And so let's see some examples of transitive relations. Well, equality is a transitive relation. Less than is transitive, greater than is transitive. Less than or equal to is transitive, greater than or equal to is transitive. What else? We have set containment. It is transitive. We have divisibility. That is a transitive relation. I wanna come back to this divisibility one for just a second. Actually, I wanna focus on this one for a second, but actually because with regard to symmetry, there's something important to mention here. Is this a symmetric relationship? Well, it depends on the set that you're considering. If you think of divisibility equipped with the integers, so if you take integers with divisibility, it turns out this is not symmetric. Well, it's not symmetric. You actually never get symmetry here. I meant to say anti-symmetry. It's not anti-symmetric. It's not symmetric because like two divides four, but four doesn't divide two. So it's not symmetric. But is it anti-symmetric? It's also not anti-symmetric because you get two divides negative two. You get that negative two divides two, but you don't have that two equals negative two. So with regard to the integers, divisibility is not anti-symmetric. It's not symmetric. But if you change the set to the natural numbers, so you've looked at the natural numbers with respect to divisibility, it's still not symmetric, but it is now anti-symmetric. In that situation, if A divides B and if B divides A, then we actually have that A equals B. And for this reason, because divisibility is anti-symmetric and we restrict ourselves to the positive integers and zero, this is oftentimes why when we talk about divisibility, we restrict our attention to natural numbers or just positive numbers if we don't want to include zero either. Because we want anti-symmetry for this relation, we just had to introduce those terms yet. So divisibility is an anti-symmetric relation when you're on the natural numbers, but not with the integers. So the set in play does matter. It makes a big difference there. I should also mention with regard to this symbol, not that symbol. I don't want to make it, you think that it's transitive there, but with regard to symmetry, right? Is it possible that A is an element of B and B is an element of A? With sets, again, we don't allow this thing to happen. No, no, no, no, no, no. In this case, you cannot have A as an element of B and B as an element of A. So that never happens with sets whatsoever. And therefore that actually makes this set symbol, this actually makes it anti-symmetric. Because the hypothesis of the conditional never is satisfied, it's actually vacuously true that this is an anti-symmetric property there. It's not symmetric, though, for sure. So those are some of the things I should have said on the previous slide. Anyways, let's get back to transitivity. If whenever R is related to Y and Y is related to Z, it must also hold that X is related to Z. So equality less than, greater than, less than or equal to greater than equal to set containment divisibility. These are all examples of transitive relations that we have seen previously in our lecture series. With regard to the relational digraph, you'll notice that a relation is transitive if whenever there's a path from one point to another point, that actually implies there's a direct arrow. So look at this graph right here, which represents a relation. There is a path from, say, five to one. Notice you can take this path from five to one. Since there's a direct path, that's a good sign there. There is a path from, say, five to two, but there's also a direct path in that situation. And so a graph is transitive exactly when there's always a direct link whenever a path exists. So this is a transitive relation. This is also a transitive relation. When you look at this one, this one is also transitive. Whenever two points are connected, there is a link that actually connects both of them there. Let's look at a few other ones. This is an example of a transitive relation. Again, whenever there is a connection between two points, there's always a direct path between them. Is this one transitive? Okay. You'll notice that four is all by itself. So it's disconnected from the rest of the group. That's okay. That's perfectly okay. To be transitive, what has to happen is that if there is a connection, there's a direct connection. So notice in this situation that there is a path from one to five, but there is no direct path here. So this is an example of a non-transitive relation. So it is possible for relations to be non-transitive. All right. Some other ones I should talk about. Let's talk about like approximately. It's approximately a transitive relation. Well, what does approximately actually mean? If two numbers are approximately equal to each other, it means they're close to each other. And so you can have it like, you can have a number X1 that's approximately X2. You can have a number that's like X2 that's approximately X3. You can have a number that's approximately, X3 that's approximately X4. But the thing is approximately doesn't mean like, it just means they're close to each other. So if one and two are close and two and three are close and three and four are close, eventually there could reach a point where you have like somewhere down this sequence, you have like X in minus one is approximately X in, but you have that X1 is not approximately equal to X in. Like, cause the thing is like, this means they're close and these are close. But when you put them together, even though like X2 is close to one and close to three, it could be that X1 and X3 are not close enough to be considered approximate, right? And so approximately, honestly, the approximate relation is not really well-defined. What does it mean? Like why is pi approximately 3.14? Like what does that actually mean? It's a little ill-defined, but if you take that idea of closeness, we can make this rigorous and calculus time on epsilon delta. But there could be a point that just because these two are close and these two are close, it doesn't mean that these two are close to each other. Okay? So this approximate symbol is, it's symmetric, it's reflexive, but it's not transitive. Okay? And so these topics we're introducing, it's important to realize they all offer something new that you can have a relation that's not transitive but is symmetric-reflexive. You can have something that's reflexive and transitive but not symmetric, like less than is such an example. It's less than or equal to it's reflexive, but transitive but not symmetric. It's actually anti-symmetric. But you can take equal, for example, equals reflexive and transitive, but not anti-symmetric. It's actually symmetric in that situation. So you can construct examples of relations that satisfy some of these conditions, but not all of the conditions, which is why we've introduced all five of them. I also wanna mention that this symbol, this set element is also an example of a non-transitive relation. So you could have things like A is inside of B and B is inside of C, but we definitely should not have that A is inside of C. That'd be kind of weird. Maybe, maybe not, it depends. It actually can happen sometimes, but let me show you it's not transitive. Let's take, for example, we'll take the set A to be the set that contains the natural numbers, the integers, and the rational numbers, something like this. And so notice that zero is an element of the natural numbers. And the natural numbers is an element of A, but we have that zero is not an element of A. So again, this shows that it's not transitive. And so these five properties are gonna be properties of relations we explore in the future. They're all independent of each other. A relation can have some of the properties, but not other ones. And they're not necessarily mutually exclusive. There are some times where even opposite looking properties can be cohabitating, like you can be reflexive and irreflexive in one special case. You can be symmetric and anti-symmetric in some special cases and such. Definitely you can have some of these properties together. And we're gonna talk about relations in the future that have some of these properties and then develop the theories around those.


Okay, this means we finished early so we can take some more questions that we couldn't take before so we have 10 minutes for postponed questions if people can still remember their original questions Yeah, and we have a Whose mouse was that by the way it can be return. Okay. Yeah Okay, so any questions that was still postponed. Okay. Yeah, I had a question regarding the GPRS and Channels and how many Basically, if if a user is connected via GPRS, does it use up that traffic channel? And how long does it use it up and how many users can use that that traffic channel? He's sorry. Could you repeat it? Okay, I'll answer sorry so The allocation is dynamic on GPRS. It's not unlike circuit switched where basically the the Each call takes up one time slot whether it's a full rate or a half rate time slot In GPRS basically that even inside a single time slot the allocation is dynamic So basically if there is data for 10 phones, then it will Take some some messages are sent for one phone some messages and for another phone. So it's completely dynamic Like you would have in any other packet oriented shared media, let's say Wi-Fi or something like that. Okay Well, there you have collisions. So it doesn't really match that way, but okay. Anyway, it's dynamically allocated So there is no theoretical limit on how many Phones can use that channel at the same time. Of course at some point it will get infinitely slow So that's a practical limit, but theoretically you can have you know as many phones as you want And they can exchange data over concurrently even over single a time slot or over collection of time slots There is one limitation though in the number of temporary flow identifiers because each phone that currently actively transmits an IP packet has What's called this TBF the temporary block flow and I think it's 16 is the maximum But I'm not 32. Does anyone know exactly? I don't know exactly how many but can be like 16 temporary 64 even okay. Yeah, thank you So you can have 64 concurrently on a protocol level in parallel But then as soon as one temporary block flow has ended, let's say the the the website has been downloaded Like the current page your opening has been downloaded completely and the user is just reading Then this temporary block flow is closed and the identifier is freed again. So another subscriber can use it So it's very Dynamic and I wasn't asked but following on this. So this is on a protocol level, but specifically on an implementation level how does How does the osmo verse handle a fair resource allocation as far as as far as gprs and edge I Don't think we fair. I don't even think we do. I mean, I don't even think we do much effort I'm not sure whether Holger has something to say about that Do you have something to say about it? Very little okay, I mean the point with gprs is right We had very very very limited resources to implement something as complex as gprs So we are happy that we have something that works at all right and and fair allocation and other esoteric topics Which of course in reality matter, but that's sort of where you know who who has an interest in in donating resources to that I mean, we're always very happy to you have okay Sure Specifically in a way that doesn't make it seem like it deserves an apology, which it doesn't we are all happy to have what we have Or we should put our money where our mouth is But I'll be more specific say for example you create a cell that has You know one whatever the required time slot is on tea at like the first tea is zero and then one through seven are all PDCH different You know MS hardware can support different like channels and block allocations or whatever else Do you max it out every I suppose I could answer this question by looking into the code But it's more interesting in the form of a discussion Yeah, look at you code now. It's it is complicated in in many ways So there are different MS classes that have different requirements how quickly they can switch from receive to transmit and You have different amount of channel combinations available Like seven slots in a row or some booth holds and what we try to do is Spread allocations equally across the PDCH. So even if you have a mobile station That is assigned to multiple time slots There's always one control time slot where the TFI is actually being used and we try to Allocate MS across different time slots equally But depending on the channel combinations You have it already limits of how many time slots you can allocate So if you're the first time slot already has some users you try to use the second one But then depending on the MS class you can only allocate two time slots and ten instead of three So this is where fairness is already Failing but the only other part is that what Daniel mentioned earlier is that you have like an allocation algorithm And he mentioned dynamic and the dynamic part is that if the cell is really really crowded We go from like multi-slot allocations to single slot allocations On the fly depending on on the loads here. Thank you I have to continue. I'm do we have more questions Three more minutes three more questions that keys Did I okay, I had a question about the femtosell and I Believe that the that one that you have sourced on that you were using it There isn't anything particularly special about it, right? It's just in a standard mode and you can access it But is there Are there any considerations in like which models there was there was something about one that was opened, right? That was what was what was good? I was you consider consideration is for you What femtosell you can actually get Okay, that's really or how much you're willing to pay because easy market is very limited They tried to have like a huge market for femtosells Which means the company's having a business model femtosells wanted to sell them in 100,000 units so mostly the operator customers tried to buy them and there's not really an open market for 3g femtosells Which was the one that that some firmware got accidentally Many I can refer to Kevin if he wants to give her a small overview So the very first femtosell which got somehow hacked was by THC. It's the voodafone UK femtosell But There's not a lot of documentation except from from this group. The second one is the one I worked on It was the SFR femtosell which was ubiquitous then Broadcom Then pico chip or all the way around They also have a 3g a newer model now, which isn't broken. The older model is still broken Then there is the one from Huawei You can access your broken broken means the security we found the security weakness so so we can access it There is one from Huawei But which we played quite of some some time with but each model We both had complete different configurations. Sometimes you art was there sometimes JPEG was there sometimes the The firmware was completely locked. Sometimes the symbols were stripped. So we stopped with it How I'd also spoke about another one where we didn't get any provision This is the exact same model then the one from the newer one from SFR, which has no where no weakness has been found in a slow time and For 3g, I think that's that's the big overview except now the one which is used now by By Osmo comm and so for the accelerate 3g. This is the one which is easiest to use because You can still find it. You don't have to get through to an operator Because previously you had to get to voodafone UK SFR France. So you need a subscription, which is quite a hassle and Yeah, they are the tools and it is open. So I think the best bet would be try to found whatever they are using One more minute another question No, sorry, it's nothing One one of the issue of using the regulated frequencies, of course the legal aspect of it in moving from BTS to femtocell do we soften that issue or not? Do we suffer from that issue? Soften? No, not really because I mean with GSM depending on the jurisdiction you're in depending on the country it's still possible to get test licenses and More or less much more easily because it's only 270 kilohertz and it's more likely that somebody has 270 Hertz 270 kilohertz of free spectrum somewhere that you can use as opposed to 3g where you need 5 megahertz of spectrum Which is a very large chunk which is much less likely that anyone has a spare 5 megahertz somewhere that you can use At a given location. So that doesn't really improve the situation. I mean, maybe on obscure bands But then you would need hardware that supports obscure bands Which makes your choice of femtocells even much less limited unless you go for like one of those software defined small cells Which have no band filters and so on and they're completely reconfigurable, but then what phones do you find for the obscure band? So it's Doesn't really improve the situation


Hi everyone, Joe for Jasper's case breaks calm coming at you with 2021 Bowman inception baseball 16 box full case pick your team number eight So I guess yeah, some teams won't have autos, but those teams are cheap promo entry spots Remember we're gonna give away some break credit at the end of this if you follow the instructions there Big thanks to there's the case big. Thanks to Everybody here for making this happen appreciate it here on Sunday the 20th 2022 Pick your team number eight Thanks everybody for making this happen appreciate it. It's like dusty got last spot mojo with the Astros There you go Good luck So that's the case of inception right says 21 boom bow inception on it. Let's see what we got Empty box and you can see the rest of the boxes The top half of that case on the top camera right there 2021 Bowman inception. So Two autographs per box It's Bowman prospect hunting here. There should be a couple cars and then a couple of autos I think that's how it goes Let's use that blank card to slow play some of the hits or so play the autographs And obviously everything ships here. That's 40 out of 50 Tyler Soderstrom for the A's That'll be for Patrick and your autographs we have we're gonna lead off with Nick Bitsco for the Tampa Bay Rays 392 out of 600 that'll be for Guy and the Tampa Bay Rays and then there's David Calabrese 59 as 75 for the halos Patrick Davis with Los Angeles with the Los Angeles California Anaheim Angels of Orange County. Oh, I just noticed that Rex saying Jason got you Silver pack one wander front ball on Friday. Nice 72 out of 99 Ed Howard for the Cubbies that'll be for Charles Three autographs. What are we what's going on here? We got AJ Vusevich Vukovic 401 out of 600 That'll be for Dusty and the dime back to one of his last spot mojo teams Last spot mojo 70% they so they say 70% of the time hits a hundred percent of the time Then we got a another autograph. That's Nick York 24 out of 50 for the Red Sox that's gonna go to Ethan and back here. We got Trey Cruz Five 64 out of 600 for the Tigers Charles Charles with Detroit I've not seen the autograph of Benny Montgomery He's doing all of his autographs differently. Some of them. He's drawing an airplane and saying But but Benny for Benny and the Jets All right, interesting. It's got to be his walk-up music. Oh a plate. All right, Pete It's Jalen Palmer one-of-one at printing plate for Must be for the Mets right there. Yeah, there's the Mets logo right there And that's for Charles and the Mets Charles all aboard the big hit Express And we got Xavier Warren 281 out of 600 For the Brew Crew that's gonna be for Brittany and the Brewers And we got Joelke Cespidus 15 out of 75 I always like this parallel because it's called the Primordial prospect and that's the that's the primordial soup in which these prospects evolve I get it. I see what they did there That's for the White Sox. That's Uens's half-brother. That's for Eric the Chicago White Sox Next box All right, we got Luis Rodriguez for my Dodgers 84 out of 99 Goes to Dusty picked up my Dodgers And we got Austin Martin One of the big prospects for the Minnesota twin that'll be for Patrick First autograph is Ezekiel Tovar 40 out of 600 Colorado Charles and we've got Jonathan India Nope, Johan Rojas 12 out of 50 Just making up names now Phillies that's for Eric I saw Jay on a Jay and a red team and I got excited Phillies that's for Eric C Next box How much of these cards run as in what what do these go for on the secondary market or what? Does each box cost Nolan Jones? Etc etc and we got Evan car Evan Carr. It's kind of hard to read these Names with the reflection out here 69 out of 600 For the Rangers that's going to be for Ethan I've got Reginald Presciato 24 out of 99 for the Cubs It's going to go to Charles and the Cubbies And back here is an angel. We got Deshaun Knowles 526 out of 600 Another one for the Halos that'll be for Patrick Davis Logan wants next year's inception to include Current players as well Well, I think Bowman probably won't do that But I guess what you're asking is you want to see Tox inception, right? Got a redemption down there And a couple others And we've got uh, Isaiah Green for the Cleveland Indians 156 out of 600 This is for you Patrick with that one And the redemption Congratulations to somebody You are due to receive a inception silver signings of s E Seattle Julio Rodriguez nice One of the big prospects for Seattle Charles with the m's A free top load This doesn't expire until October 2023. So you got plenty of time To redeem that there's a silver signings down there. All right All right 29 out of 50 hunter bishop for my rivals the Giants It's going to go to dusty and then we've got another one for the Giants Hyro a pomaris 216 out of 600 that's for dusty and the Giants And for the Nats, that's Armando Cruz 75 out of 99 silver signature Going to Cory and the Nationals. There you go little new guy mojo All right next box Grayson Rodriguez to 99 for the Red Sox we got Miguel Belize Belize 35 out of 75 Miguel going to Ethan and Boston And we've got a Pedro Martinez not that one But Ray's prospect Pedro Martinez 131 out of 600 And that's for guy Tampa Bay Is that right then? MJ Michael Jordan challenge magic to a game of one-on-one. When's that happening? I didn't watch that What could I accidentally grab an auto? There's Tanner Burns 272 out of 600 for Cleveland Patrick Cleveland, this is for you It's an Austin Wells And we've got another Cleveland player on Hill Martinez 166 out of 600 Cleveland, this is for you And back here is an angel to Sean Knowles again 69 out of 99 Another halo for Patrick There so there's a video of Michael Jordan on Twitter of him laying down the challenge Let's let's make that happen. Will this be like one of the slowest games of one-on-one that we'll see they're not exactly uh Both of those guys aren't exactly spring chickens, right? All right. There's Austin Wells for the Yankees to 99 It's gonna go to dusty And then we've got a another national that's Cole Henry 502 out of 600 Cole for Corey And the Nats and we've got another angel That's Jose Bonilla 72 out of 75 for Patrick That's I think it's I don't know if it's a movie Logan. I think that's a Isn't a tv series or is it a movie? But yeah, the Showtime era Lakers getting their own movie or series On HBO max. I want to say which looks pretty good with I think John C. Riley as It's dr. Jerry bus another redemption on the bottom and we've got Nick Gonzalez for the pirates seven out of 99 Dusty with the buckos And then the redemption Congratulations to someone you are due to receive An inception silver signings of No, we see the name first. So a short city s P That's Detroit. That's Spencer Torkelson. Nice Jay, that's a ridiculous question You're just trolling now, right? I wasn't born yesterday. Jay Jay's asking who's a better shooter. Steph courier LeBron All right, there's Pete Crow Armstrong tubs to 75 That'll be for Charles There's Asa Lacey two out of 50 That's for Michael L and Kansas City Right, bam. Neither. It's Ben Simmons 192 out of 600. That's Johan Rojas again for the Phillies Rex is saying seems like they could still do spring training for the minor league prospects Yes, I think that was always the the plan the minor league season can Obviously still go on without a collective bargain agreement in place. All right. There's Zach fiend 52 out of 99 Carlos Rodriguez 14 out of 25 Carlos Rodriguez for the brew crew, Brittany with the Brewers There's Antonio Gomez Yankees 306 out of 600 for Dusty and the Yankees Who thinks a better shooter curry or Michael Jordan? Steph Curry. I think Steph Curry is like Just as a pure shooter. I'm not saying he's better than Michael Jordan, but as a pure shooter Steph Curry is like in the top three top top of all time.  I have to think so There's uh, Brian Buelvis 217 out of 600 for the A's Patrick Davis And there's Xavier Warren another brewer for Brittany 66 out of 99 No, you're you're asking because you're you wait. I'm the expert. I'm not sure if I'm an expert, but But yeah, I think it's pretty clear that that step is a unique shooting talent No, we're not gonna do that. Bam There you go, Logan. That's the right answer All right, there's Matthew Libertor to 75 for the Cardinals And there's a Yeti cap six out of 50 There's Yeti cap for the Marlins that'll be for Dusty And the Matthew will go to Cory and the cards There's Jefferson Cuero 262 out of 600 I don't know what he's doing with his autograph. I feel like he's trying to make like a little Little cartoon character or something like that Brew crew that's gonna go to Brittany And there is my sale Rubina seven out of 99 Final shot one second on the clock. Who am I taking? Michael Jordan, but that's two different questions That's clutch shot versus shooter. Oh, and um, my sale Rubina is a twin that's gonna go to Patrick I don't know if I said that there you go Patrick All right, final box Good luck everybody. Thanks for getting into bowman inception baseball There's Michael Toglia to 75 and 59 out of 99 Nick York For the Red Sox that's gonna go to Ethan last but not least The autograph is for the Reds the Red Sox sweeping the box Gilberto Jimenez 89 out of 600 It's almost like a personal box for Ethan with both autos There you go gang. That was a 16 box pick your team number eight from jazzy's case breaks.com Quick little recap here. Thanks everybody for making this happen Appreciate it Nice Spencer Torples in there. Love all these on-card autos very sharp design And uh, some good some good prospect hunting here and hopefully fingers crossed at all these people all these players will uh Will become future stars That's the hope anyway. Thanks very much everybody I'm Joe for jazzy's case breaks.com and I will see you next time for the next break. Bye. Bye


Personal finance practice problem using Excel health insurance payment comparison prepare to get financially fit by practicing personal finance. Here we are in our Excel worksheet. If you don't have access to the Excel worksheet, that's okay because we'll basically build this from a blank sheet. If you do have access, there's three tabs down below. There's the example tab, practice tab, the blank tab, the example tab in essence being an answer key. Let's take a look at it now. Information's on the left-hand side. I'm going to be comparing two policies and the costs related to them for doctor's visits. For example, the second tab will be a pre-formatted worksheet so you can work through the practice problem with less Excel formatting. The third tab will in essence be a blank tab. We will work the problem basically from scratch. If you don't have this worksheet, then you could just put this data on the left-hand side. I would start out by selecting the entire worksheet triangle up top, right-clicking on the worksheet, and then formatting the sales to put your baseline formatting down and I would go to currency and then put the brackets and no dollar sign and no decimal. That's the starting point. I'm not going to say okay, but just X out of it here because I've already done that. Enter the data on the left-hand side, which is the best practice that you want to do anyways so that you can practice pulling your information from the data on the left, make the sales, whatever reference they need to be, changing it, for example, to percent in this case and then make the skinny C column and then we're ready to go. So information says we got the health insurance. We've got one policy pays percent of physical therapy costs after deductible of 70%. The deductible is $400. So we got a deductible of $400. You got to clear that and then they're going to be paying the 70% after that point. We have an HMO policy where there's a cost per visit for physical therapy, which is just a flat $30. That's how much we can expect to pay. And then the number of visits then we're going to assume to be 80 and the cost of the actual physical therapist is charging is $80. So in other words, with the first policy up here, we would have to split on the 70, 30 on the payments of that $80. And then down here, we're going to just have the set fee of $30 and then the difference being paid up to the 80 by the insurance company. So let's just do some comparisons here with this data. We're going to say let's make the health. This is going to be the health insurance insurance policy personal costs. So what's going to be the cost to us or like the out of pocket cost? In other words, for this first item up here, where we have the deductible, I'm going to make this sell a little bit longer, put in my cursor between the D and the E left click and drag into the right. And that's good. And let's make this one black and white. I'm going to go from D1 to F1. And then let's make that black and white as has been our custom home tab font group. And we're going to go to the bucket and make it black and then the letters and make it white black and white. And so then I'm going to say let's calculate the cost. So I'm going to say the cost is going to be the number of visits times the cost per visit. So the number of visits, I'm going to say this equals and I'll put this in our data down below so that we can pull it from the data down below. And we can run different scenarios once we get that once we get our information set up in our table. So that's going to be 12 visits, we're going to say, and then we're going to say that it costs $80 per visit $80 per visit. So we'll scroll on down here to the $80 per visit. So there we have that. So that's going to give us our total cost. Let's call it total cost. The total cost is going to be equal to I'm going to put this in the outer column equal to I'm going to say up to the 12 times the 80 12 times 80 and enter. Let's put an underline here and say let's do an underline and let's do some fancy formatting. I'm going to select these three cells for the fancy formatting. We're going to go to the home tab alignment indent and then I'll indent this double time home tab alignment indent again. And then we're going to say, OK, so then we have a deductible. So we got to clear the deductible before the insurance starts kicking in on this first plan. At least we're going to say the deductible is is $400. So we're going to say $400 on the deduct. And so then we got the cost less the deductible cost less deductible or the amount over the deductible here that we have. This is going to be equal to the 960 minus the $400. So that means we got the 560 here. Let's put an underline under this one and let's go to the home tab font group and underline to do it. And then we've got the percent percent paid by individual individual. So let's do it this way. So we're going to say 70% was paid by the insurance company of the amount over the deductible one minus 70% or 30%. Let's do a quick little calculation. One or 100% minus the percent paid by the insurance company percent paid by company is going to be the 70%, which I'm going to pull from the data equals the 70% going to percentify that cell. So I'm going to go back on it here. I'm going to go to the home tab. We're going to go to the number group and make it a percent percentifying it. And then I'm going to go to the font group and underline it as well. And this will give us the percent paid by the individual. Let's get just keep that. And I'm just going to get rid of the colon on the end tab tab. And so I'll do the subtraction problem. That one of course is 100% or one. If we made it a percent, it would be 100. This equals one or 100% minus the 70%. And if I percentize that cell by going to the home tab number group, percentification, find it, percentify it. And then we're going to say that let's put an underline on that too. While we're here, home tab font group and underline. Then we've got the and let's do some indentations here. Hold on a second. Let's do some indentations. Let's take these three home tab alignment indent and then indent this one again. Home tab alignment indent double time. And so then we got the personal cost after deductible. So that means we have this amount that cleared the deductible and the insurance company is going to pay 70%. That means we're going to pay 30%. So the amount that we're going to pay is equal to the 560 times the 30%. So that's the 168. Then we also have the deductible that we have to deal with that we have to pay. So I'm going to we also pay the deductible. So that's going to be equal to the 400, the 400, which I could pick up here or there. So the total out of pocket. So the personal costs costs including deductible is equals the s to the u to the m of those two numbers. And that gives us the 568. Now let's make it a little bit more fancy. I'm going to put an underline here and let's we could imagine imagine for example, if you will, that this number was less than the deductible. So then I could say, well, what would happen? Let's say if if we didn't visit like 12 times, but let's make this. I'm going to make this yellow and I'm going to play with it. Let's say, okay, we had it at 12. There's a percent on that. We had it at 12. Let's say that we only went like three times. Then the 80 times three is less than the $400 deductible and my whole math has kind of messed up because it makes a negative number. I don't want it to go negative here. So what I'm going to do here is I'm going to use an if then function. I'm just going to practice our formatting to say if it goes below zero, don't go below zero. So I'll do it this way equals if brackets this number minus this number is less than zero. Then that's a comma. So I'm going to say comma. That's what I say when I say then comma. Then what do you want to do if well, if it's less than zero, I want you to put a zero because I don't want you going negative. Don't go negative people. It's bad. It's bad idea. Comma and then we got the false value. If so, that's going to be this minus this meaning if it's positive, I want you to do the subtraction problem. So there we have it. Let's close up the brackets and enter. So there we have that. So that that looks good. Cost less deductible. You know, non negative. I could adjust the wording, but I'll keep it at that. And in this one, notice down here, I have the deductible. So but really, it should be amount caught paid up to deductible. So something like amount paid up to deductible or something like that, because if it's less than the deductible, I'm not going to pay the whole deductible. I'm going to pay the lesser of these two. So let's do the equals the men here equals the men of those two. And so the men is just like the sum function, but with a with a men. And so and that'll pick up the smaller number. So now I can now I can play with it. If I'm below the deductible or going above the deductible and I can start to say, okay, what if this was five, right? What if this was six? So now I'm over and so on. And if I go back up to 12 and we should have something that helps us to work different scenarios now, which is nice. I'm going to delete this. Let's put some fancy formatting around this making it blue and bordered. I'm going to select this whole thing and border blue home tab font group border and then the bucket. I'm going to use that blue. But if you don't have that, you could go to the more colors more colors. We're in the standard item here. We're going to go to that blue right there. That's the one we want. That's the one I'm using. At least you could use whatever you want. It's not like a standard rule here that's etched in a stone somewhere and even if it was etched in a stone, that still doesn't mean you have to do it just because someone etched it in a stone, right? I mean, except maybe some things etched in stone are important. But okay, home tab. Let's go make a skinny column over here. I'm going to select this one home tab and then we're going to say this is paint brushy and we'll put that over here on column G. We got a skinny G. So this is the HMO now HMO personal cost. So now we'll do this calculation for this policy down here where we're just going to pay a flat $30 and we don't even have the deductible thing to deal with. So I'm just going to say, alright, let's make this one a little bit wider. That should be easy to do and let's do our header thing selecting these two columns or rows or cells. I should say home tab font group. Let's make this black and white for the header as has been our custom white on the lettering. And then this is going to be the cost per visit. I'll just call it that. And this one is just a flat $30. That's what they tell us. It's going to cost you $30. But how much do they charge? I don't know. They just put that I got to pay $30, but they're actually charging $80, but we're paying the 30. And then we're just going to take the number of the visits, the number of the visits. And this is because they have a deal and the HMO with of course the HMO and the network and whatnot. And that's going to be 12. So we're going to go up top and say font group and underline. And so this is going to be the personal costs this way which will simply be equal to the 30 times the 12 like so and let's make that blue and bordered. I'm going to put some fancieness around it and we're going to go to the home tab font group blue and then the borders blue borders. That's what we like to see. And so the difference between the two the difference is we got the 568 over here minus the 360. So we're saving the 203 with the HMO. Now note that the HMO the idea would be that they might be able to have lower costs because you're in the network but you might be more restricted because you have the network and you've got to be within that network and and so on. So we've talked about the pros and cons of a more flexible plan possibly that you'd have to measure that you might be able to travel more or something like that and not have to worry about so much maybe a network kind of issue versus an HMO which might be a little bit more restrictive and so on. But that's the and so and then of course you'd have to compare the premiums between the two how much you know the policies are costing to kind of make a decision if you're making a decision between the two but just to get an idea of how you might set up a worksheet. You can look at your past history and try to think about okay how much am I paying on this kind of stuff to get some bearing on a reasonable decision. Once set up then we can we can increase or decrease these numbers right I could say well let's do this 12 thing again. What if that 12 changed to like what if it was like for so then so then over here now we're under the deductible but we're still paying more because we're paying the whole $80 here we're over here we're only paying the $30 per visit we don't have to deal with the deductible. If we go then to six you know now we're over the deductible if we went back up to let's say the 12 the 12 so that's where we were before and if I increased it to some point if I increased it to like 50 or something like that now this one is higher than this one and you can find that kind of break even point if you wanted to play with this kind of thing you could say what number would this have to be to get these two numbers in essence to be the same right how many number of visits would I have to have to get that number to be the same to do a calculation like that you could go using the data tool over here and we could go say I'm going to click off of the cell I'm not going to go over here to the what if analysis and goal seek so I could say seek a goal what do I want to seek what's the goal that you are looking for the holy grail okay anyways we're going to set the cell we want to set this cell to know we want to set this cell to be zero that's what we're looking for and then we want to do that by changing this cell so we're saying Excel change this number to whatever it needs to be to make the difference zero using trial and error goal seek okay and so so there we have it is forty seven oh man what happened I just got the answer and then I deleted it let's do it again do it again we're going to goal seek so we're going to say set this cell to be zero by changing this cell poor father or don't mess it up okay so there did like a forty seven right so you can do something like that and you can also change of course this one if I bring this one back to twelve let's make this on yellow on yellow fire and now I changed this one and say okay what if my what if the doctor's costs go up to like you know forty like what if they were only what if they were only forty dollars on a doctor's cost right then you got a smaller difference with their fifty you know what if they the doctor's cost go up to a hundred and twenty or something like that and you can again change your your data and run run basically different scenarios based on that if you have your worksheet set up which can be useful so let's bring it back to where it was where was it I wrote down twelve I was what was it before what was it people okay eighty that's right thank you okay so there we have it let's do a spell check real quick does miss spell anything of course you did of course you did I'm just going to say alright that's good we'll keep it at that


Good morning. Hello, Zambia. How are you? My name is Kaye Felix And I just want to wish all of you a happy happy independence day. I love you Bye-bye