 Hello, and welcome to this first video for chapter two in the Sundstrom textbook, and in this video we're going to work on the negations of simple statements in English. So in one sense it's very easy to form negations because you just stick the word not in front of whatever statement you're trying to negate. But there's better ways to do this usually that involve better phrasing and communicating in mathematics. We're all about the phrasing here. So I have three examples we're going to work through here. First of which, let's take the statement that the number 18 is prime. Now this is actually a false statement, but at least it's a statement that has a truth value. And we're going to, in each of these examples, we're going to write down a basic form of the negation, and then a better form of the negation that we should probably shoot for. So the basic form of the negation for this statement is just to simply say that the number 18 is not prime. The number 18 is not prime, you know, to stick in the word not in the appropriate location of the sentence. That is certainly a correct negation, but there's actually a slightly better way to say this and involve some new terminology. In mathematics, we often say that a number that is not prime is what we call composite. That's a technical term. You may not have heard that ever until just now, but now you know. So a better way possibly to say this is to say that the number 18 is composite. Number 18 is composite. That is one way to phrase that negation. It avoids the word not, because sometimes it gets in the way. Second example, the number 36 is even. Of course, the basic way to negate the statements is to say the number 36 is not even. Again, that's correct, but it's a little awkward because we have another way to say what we mean when something is not even, and that is to say that the number is odd. So it would be better to say that the number, the number, again, my handwriting showing its skill right now, the number 36 is odd. Now that, of course, is a false statement, but we're not getting into the truth and false set of these statements right now. Just how to phrase their negations. Okay, so instead of saying not even, let's say odd instead. It's a little bit more to the point. Lastly, and this one is slightly tricky, the statement 10 is less than 8. That's a strict less than here. Now the basic phrasing of this to say it in English is to say that 10 is again not less than 8. That would be correct, but there's a better way to say it that takes up less space and gets the point across more efficiently. Now what do you think it means to say that 10 is not less than 8? It would not be correct, and I'll put this in red because this would be an incorrect way to say this. It would be incorrect to say that the negation is that 10 is greater than 8. Can you think about why? Pause the video for a moment and just think about why is that incorrect? That seems right. So the reason this is incorrect is just because 10 is not less than 8 doesn't mean that it's greater than 8. It could be equal to 8 somehow. So the better way to say this, to negate this statement, is to say that 10 is greater than or equal to 8. So there's three simple examples about how to negate a simple statement. In later videos we'll be moving on to using these negations to do bigger and better things. So thanks for watching.