 Hello, and welcome to the screencast on conditional statements. So in the last video, we looked at a particular statement about prime numbers, namely that statement that for every prime number, p, or whenever p is a prime number, the number two to the p minus one is also prime. Now another way to say that is to use the words if and then. We could say it like this, if p is a prime number, then two to the p minus one is a prime number. Now any statement that has this form or can be rewritten in this form if something, then something else is called a conditional statement because it's saying that if the condition is met, then something automatically happens. Technically we say that the statement following the if the condition is called the hypothesis and the statement following the then is the conclusion. A conditional statement is a claim or a promise that whenever the condition or hypothesis is met or satisfied, then the conclusion automatically happens. So this is one example of a conditional statement and the hypothesis is p is a prime number. This is the condition under which something is supposedly guaranteed to happen. And the conclusion, which is the thing that happens, is two to the p minus one is a prime number. Now in this particular case, we saw that if p is a prime number, then it's actually not always the case that two to the p minus one is prime. This tells us something about whether our claim or our promise is really honest and we'll get to that later. Now life is full of conditional statements. I have three little kids at home and their whole day is made up of conditional statements like if you finish your dinner, you can have dessert. Or if you spray your sister one more time with that whipped cream, you're going to your room. The hypothesis of the first statement is you finish your dinner and the conclusion is you can have dessert. Note that there's no then actually written before the conclusion but it is implied. The hypothesis of the second statement is you spray your sister one more time with this whipped cream and the conclusion is you're going to your room. Now these are promises I make. My kids understand, or at least I hope that they do, that whenever the hypothesis happens, the conclusion automatically happens and this helps them make decisions. Here are some more examples. If it's not raining outside, then it's not cloudy. The hypothesis is it's not raining outside and the conclusion is it's not cloudy. Here's another. Every time x is a rational number, x squared is also a rational number. Now you don't have to understand what a rational number is to understand the statement. Here we don't see the words if or then in the sentence but as in the previous example, they're implied. The every time part tells you where the hypothesis is. That's just another way of saying if, so the hypothesis here is that x is a rational number. And the conclusion follows the hypothesis. So the conclusion here is x squared is a rational number. Here's another one that's a little more complex. And although it's complex, we're going to do this as a concept check. So as with all the concept checks, listen to the question and the responses, and then pause the video while you decide which response is most correct. Here's a conditional statement. If it is not raining outside and the temperature is above 70, then I will go to the beach. So the question is, what's the hypothesis of this statement? Is it it's raining outside? It's not raining outside. The temperature is above 70. It's not raining outside and the temperature is above 70. Or I will go to the beach. So go ahead and pause the video and think of which response is most correct and unpause when you're ready to go ahead. So the most correct answer here is D. It's not raining outside and the temperature is above 70. Now you may be thinking this is not one hypothesis but two hypotheses. But in fact, this one whole statement is one claim. This is the condition, literally the weather conditions, that if the conditions are met, the condition is met, will trigger me going to the beach. And as my kids can attest what I've made promises about going to the beach like this, if the conditions are met, but we don't go to the beach, then there was something dishonest about my promise in the first place. And I will definitely hear about it from them. We'll have more to say about and statements later on, which is what this hypothesis is. But for now, notice that the hypothesis here has two parts and they are joined by the word and. We still consider this to be a single statement in the hypothesis. If only one part is met and the other isn't, for example, if it's not raining outside, but the temperature is not above 70, then there's no guarantee that we'll go to the beach. I might and I might not. Maybe I'll decide to go anyway and just look at the sunset, even though it's 30 degrees outside. It doesn't make me dishonest for making this promise in the first place. Only when the entire condition is met is the conclusion triggered and it's only on that basis that we can talk about the honesty or dishonesty, or shall we say the truth or falsehood of the original statement. So let's wrap up what we've learned here. First, a conditional statement is a proposition of the form if A, then B, where A and B are individual statements of their own. The statement A is called the hypothesis of the conditional statement. And the statement B is the conclusion. And the way conditional statements work is that when the hypothesis condition is satisfied, anytime it's satisfied, then the conclusion automatically follows. If the conditional statement is what we'll say quote unquote honest, and we'll have more to say shortly about what honest means in that context. Thanks for watching.