 Hello and welcome to the screencast on negating quantified statements. So recall from earlier work that a negation of a statement is simply another statement that has exactly the opposite truth values in all circumstances. We often use if the statement is called P then we use the symbol not P to represent its negation. So if the statement is P is even then the negation of that we could form just by putting the word not into the right place. But a better way to say P is not even is to say that P is odd. Oftentimes when we form a negation we can do it in a sort of naive way by sticking the word not in there in the right place. But it's up to us to try to find a more fluid more human readable way to state that as well. And we're going to do that now with quantified statements both existentially quantified and universally quantified statements. Those are logical statements they are either true or false and have a definite truth value. So what would be the negation of such a thing? Let's begin with an example of an existentially quantified statement there exists a prime number that is less than two. Okay so what would the negation of that statement be? Well one thing we could do is just stick the word not in the right place that there does not exist a prime number that is less than two. Or we could say prime numbers then are less than two do not exist. Okay that's fine and that's not incorrect. But maybe there's a better way to say this that isn't so negative. Can we space frame this in the positive? So what can we say about prime numbers? If prime numbers that are less than two do not exist then suppose I had a prime number say P then what can we say about it? Well it would not be less than two. It would be either equal to or it might be greater than two. And that's the gist of our negation here. One way to form the negation of this statement is just to simply put the word not in the right place. But a better way to say it is to say that all prime numbers if I have any prime number will be greater than or equal to two. That is the correct negation of this sentence up here. And notice one thing before we move to a general principle. I started with an existentially quantified statement. Existentially quantified statement. But when I negated it I didn't get an existentially quantified statement. I got a universally quantified statement. And something else changed. The property that was being stated in the original statement is itself negated. The opposite of less than two is greater than or equal to two. So when I negate an existentially quantified statement it turns into a universally quantified statement with the property negated. And we're going to say this in a sentence form here. That the negation of the statement there exists an x such that property p of x holds is for all x the property p of x does not hold. If I want to negate the statement there exists a prime number that's less than two. I'm going to say for all prime numbers the prime number is not less than two. It's greater than or equal to two. A notational way to say this is the following that the negation of there exists an x such the p of x is logical equivalent to saying for all x not p of x. So we're going to switch the universal quantifier to an exist, or the existential quantifier to universal quantifier and then negate the property that's being stated. More examples here there exists an integer whose square root is an integer. The negation of that statement would be to say for every integer k radical k is not an integer. So again we've changed from existential to universal and we're negating the property. We started with the square roots and integer. We ended with the square roots not an integer. So here's another example that's less mathematically oriented. Some math majors have red hair. At least one or there exists a math major that has red hair who has red hair. Now the negation would be to say no, no math major has red hair. Or a little bit more awkwardly phrased we're going to switch the existential to a universal quantifier. Every math major and then negate the property has non-red hair if you will. So again to negate an existential quantifier we change it to a universal quantifier with the property itself negated. So now let's turn our attention to negations of universally quantified statements. Take a look at this statement here. Every math major has red hair. Now that's a statement and we like to form its negation. Before we do that I have a proposal for a negation here but before we look at it I answer this question. Is this statement true or false that every math major has red hair? Well if you know enough people and know enough math majors and I can introduce you to some more if you want this statement is definitely false because there are a lot of math majors that do not have red hair. For example I don't have red hair and I was a math major. So it's not the case that every math major is red hair. That's a false statement. Now take a look at this statement here that you might think is the negation of the original. No math major has red hair. It's possibly reasonable to think that this is the negation of the original statement because the opposite of everybody is nobody right? Well actually this doesn't work because think about the truth value of this statement. No math major has red hair. That is also a false statement because I know math majors and I can introduce you to some again who do have red hair. So these two statements can't possibly be negations of each other because they have the same truth value. And remember a statement and its negation have the opposite truth values. So surprisingly possibly we don't negate a universally quantified statement by making yet another universally quantified statement with sort of the opposite quantifier. That just simply doesn't work. So what is the correct negation of this original statement? Well we had a little bit of a hint when we were thinking about why this statement was false in the first place. It was because we could come up with an example of a math major who does not have red hair. That's why that statement is false. The existence of that example is how we're going to get our negation. Let me just erase some of the wording here. The original statement is false because not every math major has red hair. That would be actually one way to form the negations. Just again simply to put the word not in the right place. But let's see if we can phrase that a little bit more positively. The way we know that not every math major has red hair is to say that there exists a math major. I'm going to abbreviate it a little bit. With let me say non red hair. That works. The fact that I know that there exists a math major with say brown hair means that not every major has red hair and that's definitely a true statement. That was actually the opposite of this statement. So notice when we negated the universally quantified statement it changed from a universal quantifier to an existential quantifier and again the property has reversed itself. Let's put this together and come to our conclusion. The negation of the statement for all x p of x holds is there exists an x such that p of x does not hold. And in notation language this is how we would say that. The negation of for all x p of x is there exists an x such that not p of x holds. And here's an example. Every integer is odd. Now what would be the negation of this? It's not in front of it and that would be okay but what does that even mean to say that not every integer is odd? Well what it means is that there exists an integer that is not odd and maybe a better way to say not odd would be to say even. Another example here. All functions or every function is continuous. What would be the negation of that statement? Well if not every function is continuous then there must exist a function that is not continuous or discontinuous. So we negate a universal quantifier by changing the quantifier again this time from universal to existential and then negating the property that's being satisfied. So here's a concept check. Look at the statement. If x is a rational number then radical x is a rational number. What is the negation of that statement? So look through your options and this is one you have to sort of think about a little bit because the way that the sentence is structured it's not totally clear that this pertains to the screencast but it does. So think about this as your answer. So the answer here is going to be d. There exists a rational number x such that radical x is not rational. There are a couple of ways to see this and I want to tie our new material into something we've already learned. First of all the sentence that we're working with is an if then statement. It's a conditional statement. And one of the strong points we made earlier in our study of statements and logic is that the negation of a conditional statement is not another conditional statement. So if you just remember that you can automatically wipe out a and b. Now what does this have to do with universal quantifiers? Well there's a connection here between if then statements and universal quantifiers. What we're really saying here is that when I say if x is a rational number what I'm really saying is for all rational numbers x for all x's that are rational then a property is satisfied. Namely that radical x is rational. So there's an important concept in this concept check that every if then statement can be rephrased as a universally quantified statement. So what we're really looking at here is the statement for all x and the rational numbers radical x is rational. Now to negate that we've seen now that we're going to have there exists an x and the rational such that radical x is not rational. And that is exactly what statement D says. So now we know how to negate universal and existentially quantified statements. Congratulations.