How did you get on with assignment five? Mastery of quantifiers is the one last ability you need to be able to cope with definitions of mathematical concepts and with mathematical reasoning. I'm going to begin by looking at the way negation affects quantifiers but before I start, let's see how you are progressing by way of a quiz. Okay, let's see what we have. Is it that case that for every real number x, x plus 1 is greater than or equal to x? Yes, it is. Is it true there is a real number x? So x squared plus 1 equals 3. Well, that would mean x squared equals 2, so we're really saying, is it true that the existing number x whose square is 2. But if we were talking about the rational numbers, the answer would be no because the square root of 2 is irrational. But for the real numbers, it's true, what about this one? Well, for very large negative x, x cubed is an extremely large negative number which will dominate all of these numbers. So for large negative x, this expression, this cubic expression, is negative, so it's not the case that it's always positive. So that one's false, so what about this one? Is it true that for all x there is a y? X cubed plus y cubed equals 0, that would mean y cubed is equal to minus x cubed. The answer is yes. Given any x, just let y be negative x. When you cube a number, you cube a positive number, is positive, you cube a negative number is negative. So, by taking y equals negative x in all cases, you get 0, so this one is true. What about this one? Is there an x such that whatever y you want to it you get 0? Well, the answer is no. If the quantifiers were the other way around, the answer would be yes. For every y, there is an x such that x plus y equals 0. Namely, you let x be negative y but it's not the case of is 1 x that works for y so that one is false. What about this one? Well, is it the case that for all x there is a y such that if x is a non-negative then y squared is equal to x? Well, let's look, suppose that we took an x, if x was negative then game over because this conditional would be true. It would have a false antecedent, if that x was not negative, then there is a y whose square is x, remember the square root of x. So what ever x we take? This always going to be a y that satisfies this. It's true. You might have to think about this one a little bit, Because there are two cases involved, given an x, you can find a y that makes this thing true. If the x you're given is negative, any y will make that true. Because if the x is negative, the antecedent is false. But if the actual given satisfies this condition, if the actual given is non-negative, then the y you picked is a square root of x. See what's going on? If you are given a negative x, there's no y that satisfies, y squared equals negative x. Not in the real numbers but that's a limit, it's that case, so this one's quite subtle. I suggest you think about this one, it's the kind of thing that you encounter quite a lot in mathematics. Okay, well how did you do? Okay, let's move on. In mathematics, and in everyday life, you often find yourself having to negate a statement involving quantifiers. Yeah, of course, you can do it simply by putting a negation symbol in front but that's not enough, at least often it's not enough. You need to produce a positive assertion, less a negative one. The examples I'll give, should make it clear what I mean by positive here. But roughly speaking, a positive statement is one that says what is rather what is not. In practice, a positive statement is one that contains no negation symbol, or else one in which any negation symbols. Are as far inside the statement is as possible without the resulting expression being unduly cumbersome. Let's look at our first example of negating a quantified statement, let A(x) be some property of x. For example, x is a real root of the equation x squared + 2x + 1 = 0. I'll show that not four x a of x is equivalent to exits x, so it's not 3 of x. For example, it's not the case that all motorist run red lights, is equivalent to there is a motorist who does not run red lights. Well, in this case, it's pretty obvious that these two are equivalent. The proof that I'm going to give in the general case, is actually the same reasoning you automatically do when you look at this specific case. If it seems hard to follow the issue is purely one of the abstraction. I'm proving an equivalents, so I'm going to prove an implication from left to right and then from right to left. So we begin with the left to right case, I'm going to assume not for all x A of x. Well, if it's not the case that for all x A of x then at least one x must fail to satisfy A of x. So for at least 1x, naught A(x) is true, in symbols, there exists an x, so it's the naught A(x) is true. Well, that's the left to right implication. I assumed not for elective of x and I concluded there exists an x not a over x. Now we want to do the other implication. So assume exist x not a of x. Well, in this case, there's an x to which fx is false that's just expressing this in everyday English. Well If there's an x for which A of x is false then A of x can not be true for all x. In other words, for all x, A of x must be false, expressing that in symbols. It's not the case that for all x you have x and now I've proved the second implication from exist x not a of x to not for all a of x. Assume exist x not a of x conclude not for all x a of x. If you found this reasoning hard to follow, it’s purely because of the abstraction. The logic is exactly the same as in the example, almost certain that you have no difficulty with this example and that’s because you’re familiar with the situation. The human brain is very good at doing logical reasoning about familiar situations. When we tend those situations into abstracts visions, the brain finds it difficult, at least at first. And in fact, one of the things about becoming a mathematician Is learning to take familiar everyday of reasoning that we don't even think about. Which is do it automatically and reproduce it in an abstract situation and it's the abstraction that's causing you difficulty. You can follow logic, but you find it difficult to follow it in an abstract situation. And that's just because of the where the human been x, okay. Why don't you try this one? Show that not exist in x of a of x is equivalent to 4 x, not t of x and before you start, you might want to look on everyday example. This is coming up as an example in the assignment for this lecture but you might want to try it now while this example is freshen your mind. Okay, good luck at that one with that one under our belts now we can go back and look at that aerial example of the negation of all domestic cost that are badly made. The C be the set of all cars D of x means that x is domestic, and M of x means that x is badly made. With this notation the sentence becomes 4x and C, D of x Implies m of x. For all cars, if the car is domestic, then it's badly made, we've already seen what happens when you negate a universal quantifier. The negation becomes there exists an x in c such that dx does not imply and here I've actually abbreviated the following. Not the case DX implies M of X but instead of writing not dx implies m of x, I use this simple notation dx does not implies m of x. But in any case, looking at our previous example, the one that I illustrated it with the motorist. When you check for all x something and you negate it, you get the exist x with the negation of the thing inside just compare what's going on here with the previous example. One comment, why am I not saying exists X not in C beginning students often do this kind of thing. And the reason they do it is they are trying to go through a formula symbolically and negate things. And that's not the way to go about these things you got to think about what the symbols mean. In this case, it's talking about Xs which are in c. The only x's we're interested in are x's which are in c, in other words the only object we're interested in are cars. But if the only objects we're interested in are cars the negation will only be talking about cars. So the m c part simply tells us which kinds of objects we're dealing with, and we don't negate that. Okay, now let's look at this part. We know that dx does not imply m of x, Is equivalent to, D of x and not M of x. In fact, this was the key part of our reasoning to figure out the truth table for the conditional. D of x does not imply m of x if d of x can hold and nevertheless m of x can fail, so we looked at this when we did truth tables. So the negation of the original sentence, Which is this, Is equivalent to there exists x in c This part is the same as this part, D of x and not m of x or what does this mean in everyday language. It means, there is a car which is domestic and it’s not badly made and that’s it, so if you found it difficult to figure out the negation of this sentence when we first looked at this example. Now, because we've got this notation and this little method for dealing with these negations, you should be able to follow it so fairly straightforwardly. Moreover, the previous time when we looked at this, even if you got the answer, you might not have been totally confident that you got the answer right. This kind of reasoning makes it clear that this is the right answer and that's the whole point of introducing this formal notation and this logically precise reasoning. We're no longer left not certain if we've got the exact best correct logically correct negation. Now mathematics leads us to the answer and that is the correct answer, okay.
