 Welcome back to another screencast and in this one we're going to discuss negations of conditional statements. So back in preview activity one from the Sunstrom textbook, what you found was that this logical equivalence holds that the negation of P implies Q is P and not Q. And that is the negation of a conditional statement. To form that conditional statement, the most important thing to realize is that the negation of a conditional statement is not yet another conditional statement. That's very counterintuitive and hard to grasp sometimes. The negation of a conditional statement is actually an AND a statement where we assert P but the negation of Q as well. So let's get this under our fingers by doing some examples. We're going to take these two conditional statements that we first met in the previous screencast and just form their negations. So the first one says if it's raining outside I'll carry my umbrella. Now again the negation of this statement is not going to be an if then statement, very importantly. I'm going to say it's raining outside. It's raining. And I am not carrying my umbrella. Once again I'm abbreviating in the writing because my writing is not so great and it takes time. But here's what it says in English. It's raining outside and I'm not carrying my umbrella. That's the negation of this conditional statement. Again it is not an if then statement. It's an AND statement. It's more like an assertion than a promise. And we're going to get to that a little more deeply in the next slide. Second statement says if P is a prime number in greater than two then P is odd. Now if I were going to negate that statement I would make an assertion. I would make an AND statement. I would say that P is prime and P is greater than two. And again it's not an if then statement. It's an AND statement and P is not odd. P is, in other words, even. And that would be the negation of this if then statement. Once again the most important thing to realize is that the negation of a conditional statement is not another conditional statement. Get this well understood in your mind here. Now intuitively why should we expect the negation of a conditional statement to be an AND statement? You can look at the truth tables and you did if you did preview activity one. And that's all well and good but if we don't believe it in our hearts it's not going to stick with us. So why should that be? Let's go back to a conditional statement that I use in my kids. If you finish your dinner then you can play outside. That's my original conditional statement and here is its negation. You did finish your dinner but which is another way of saying AND. You cannot play outside. Now why are these two things negations? Well if you believe this then I'm making a promise. I'm making a promise that is true. That makes me a good guy. If you believe it's negation I'm a liar. And I'm a bad guy. I'm a bad parent. And we should all be sad about that. These two things are completely opposite each other. If I say one and then do the other that's a total contradiction in terms. Literally a contradiction. These two things cannot be true at the same time. It's not a conditional statement. It's an assertion that I have broken the promise set up by a conditional statement. So again the negation of a conditional statement is not another if then statement. It's an AND statement that says I did satisfy the hypothesis but the conclusion didn't follow. So let's end this off with a concept check. Here's a statement if A and B numbers A and B are even then AB is even. Then what is its negation? Process read through, pause the video and come back when you're ready. So if you've been paying a little bit of attention to this video you know you can completely eliminate the first three. How come? Because they're if then statements. And the negation of an if then statement is not another if then statement. So that rules those three out automatically because they're if then statements. So what is the negation? I would have to, just like with my promise to my kids, I'd have to satisfy the hypothesis but negate the conclusion and join them with an AND. Both of these are joined with AND. And in D, I am saying A and B are even but AB is not even. So there is the correct answer D. In E, I have negated the hypothesis as well and that's not part of the negation of the if then statement. Thanks for watching.