 In this video I'm going to talk about verifying conjectures, verifying conjectures using the law of syllogism. Okay, so determine if each conjecture, each conclusion that we draw is valid by the law of syllogism. So we have a couple of statements here. So given, so this is what we hold to be true, given if a figure is a kite. Okay, so if a figure is a kite, so it looks like a little kite. Okay, so something like that, that's a horrible drawing of a kite. But anyway, you get the idea. If a figure is a kite, then it is a quadrilateral. Okay, so if it's a kite, then it's a quadrilateral, it's got four sides, that makes sense. So here's our first conditional statement. If a figure is a quadrilateral, then it is a polygon. If a figure is a quadrilateral, then it is a polygon. Okay, so that's our given statement. If a figure is a quadrilateral, if it's four-sided, then it is a polygon. Polygons are just shapes. The conjecture is, if a figure is a kite, then it is a polygon. Okay, so we have to go back to our law of syllogism. Law of syllogism is basically if p then q, if q then r, then p then r. So basically, if we have a logical order like this, if we have conditional statements that follow this order, this last part is going to be true. So let's see if our conditional statements actually follow that order. So if a given, if a figure is a kite, so we'll call that p, that's our first statement. If a figure is a kite, then it is a quadrilateral. Okay, so that's going to be our q, that's our second statement there. And this is a hypothesis and a conclusion, so that's our first conditional statement. That's our first part right here. All right, now the next part, if a figure is a quadrilateral, all right, so it looks like we use this again. If a figure is a quadrilateral, so q quadrilateral, so it looks like we're on the start of our second part there, then it is a polygon. So we've got a new statement here, this polygon we'll call that r. Okay, we'll call that r. So it looks like we're starting to follow that progression. If a figure p, if a figure is a kite, then it is a quadrilateral. If p then q, if a figure is a quadrilateral, then it is a polygon. If q then r. So it looks like we have followed the first two steps here, we've followed the first two conditional statements here. Now very last, let's see if we follow that same, let's see if we follow that same and get to this third step. Conjecture, if a figure is a kite, so as I look back up here, p is what I used for kite. So if a figure is a kite, then it is a polygon. What did I use for polygon, r? So did it follow our logical progression? Yes it did. If a figure is a kite, then it is a polygon. This is actually a valid statement. This is a valid statement. Now as you read through that, if you read through that the first time, you probably could have picked that up. But what I did there is I used the law of syllogism, I used these three steps, these three conditional statements, and labeled everything out so that we can visually see how the law of syllogism is used. So that's one example of using the law of syllogism to verify a conjecture. So let's do a separate example and see if we can do this one more time. So again, law of syllogism, if p then q, if q then r, and then we could also say if p then r. So we're kind of, this last statement connects the first two, p then r. So if a number is divisible by two, so we'll call that p, if a number is divisible by two, call that p, then it is even, call that q. If a number is even, if a number is even, call that q again, then it is an integer. So that's the new, so we call that r. If a number is divisible by two, then it is even. That actually makes sense. If a number is even, then it is an integer. Well that's also true, even numbers are integers. So let's come to some conjecture. If a number is an integer, which is r, if a number is an integer, which is r, then that number is divisible by two. Then that number is divisible by two. Kind of look at our progression here. Look at our logical compression. p then q, and if q then r. So we've actually followed these first two rather well, the last one here, if r then p, if r then p, it's backwards. This one is backwards. So right away we can tell that that's going to be invalid. Right away, since that's backwards, we can already tell that that's invalid. Now that's just looking at the logical progression, using these three statements up here. Now let's actually, let's think about it. Let's think logically about it. If a number is an integer, then it is divisible by two. Numbers that are divisible by two are two, four, six, eight, ten, twelve, fourteen, sixteen, eighteen. All the even numbers, they're all divisible by two. But it's saying if a number is an integer, well integers are numbers like negative three, or zero, or one, or five. The negative number is zero in these positive whole numbers. Well, right there I've already written two examples that are not divisible by two. So negative three, one, zero, five. Those are not divisible, they don't divide evenly by two. So that right there tells you that this statement, this conjecture is not valid. I just come up with a bunch of examples of where it's not valid. So that's another way to look to see if that is valid or not. So just to kind of compare, kind of recap a little bit. The first example that we had, notice that followed the logical progression, P, Q, and then Q then R, and then P then R. Followed this progression. So that was a valid statement. So one thing to do would be to label all of your different pieces, your hypothesis, and your conclusions. To label all those, that might be a little bit easier. And then over here we did the same thing. We started labeling our hypothesis and conclusions, but we noticed that once we got to our final conjecture, it was out of order. This last part was out of order, so that tells us that right away it's going to be invalid. Now if you don't follow that quite well, just think about it logically. Go through the conjecture. If a number is an integer, like these numbers up here, then they are divisible by two. Well, none of those numbers are divisible by two, so that's an invalid statement. All right, so that is just a couple of examples, a valid and an invalid example of verifying conjectures using the law of syllogism.