 The truth value of a statement often depends upon which elements of a set is the statement describing. There's a big difference between whether some students in a course are struggling versus every student in a course struggling. I of course hope for my students that none of us are struggling. If you are, please come get some help. But like, if you have like one or two students who are struggling in a class, give them help. The class is probably doing pretty good. But if everyone in the class is struggling, well, then that's a big difference, right? That's probably the teacher's fault, not the student's fault in that situation. Now the difference between these statements here, every student versus some students is this idea of quantification. I've quantified the statement and its truth value can depend upon whether it's quantified with every or for some, okay? And so there's these two types of quantifiers that come up in statements and we have to pay attention to them in mathematics and logic. The first quantifier is referred to as the universal qualifier. The universal qualifier describes things like all and every. That is the universal qualifier describes every element that belongs to a set. And the symbol we often use for the universal qualifier is an A that's upside down. The idea is you think for all and then you just write the A upside down. That means for all. In fact, the latex symbol for this symbol is actually for all. It's kind of fun there. And so let's look at some statements that have the universal qualifier in them. So statement A here, all citizens over the age of 18 have the right to vote. B here, every triangle has an interior angle sum of 180 degrees. C, each student must register for class by August 1st at some fictitious university here. But the point is these statements very much change their meaning if you say something like, oh, some citizens, some citizens over 18 have the right to vote. Well, that's not universal voting rights there. Or if some students, if some student has to register by August 1st, you're like, well, who was that student? And so I use three different synonyms here on purpose, all every each. And each of these words is used to describe the universal qualifier. Now, in contrast to the universal qualifier, there is the existential qualifier. The existential qualifier captures the idea of sum. There exist, there is at least one. That is to say that something exists, hence the term here. I mean, the names kind of means something. Universal means everything. The existential one means that something exists. Now, it could very well be that if everything has the property, then something has the property, at least one. So if you have a universal qualifier, you can always weaken it to the existential qualifier, which you typically write that with a backwards E, existential qualifier. But the other direction is not necessarily true. You could have some elements with the property, but that doesn't mean everyone has the property. And let's look at some examples of statements which have the existential qualifier. Some drivers qualify for reduced insurance rates. Maybe your insurance company is making this. Oh, some, if you have good driving, then maybe you can qualify. Some, it's not saying everyone does, because if there's any exceptions, they would make them a liar. They could get sued for such false advertisements. So some drivers suggest that, well, you could do it, right? But maybe, we're not sure, at least one exists. Or here's another one. There is a number whose square is 25. I can actually think of two integers whose number is 25, five and negative five. But the idea here is there's at least one. There could be multiple, there could be more than one, but there's at least one that exists. On the other hand, it does not guarantee that all numbers have that property. It is not true that all numbers have a square equal to 25. No, no, no, no, no. Most numbers will be something else. But there is a number whose square is 25. Or here's another one. There exists a bird that cannot fly. Penguins, ostriches typically don't fly. They're not typically known to be so. By all means, I'm not saying that all birds cannot fly. There's too many counter examples of that. But if you were to say that all birds fly, that's also a false statement because there do exist some birds that cannot fly. So you wanna be careful of your quantifiers here and see, again, these examples. Some, there is, there exists. These are all examples of existential quantifiers here. All right, so it's important that as we work with logical statements that we are able to express them using logical notation. So let's take a few true statements and then think about, what would their translation be into symbolic logic? Take for the first statement here, every integer that is not odd is even. This is a true statement and actually we could probably prove it if we wanted to. Now, notice it says every integer. This is a quantification here, every integer. So if you take an integer, an integer is an element of the set Z here. So if you say every, this is the universal quantifier there, the upside down A. So you look at this right here and you look at this right here, they say the same thing, every integer. And this says, then this right here says for all elements of the set of integers, which is the same thing as saying every integer. So we're now qualifying what elements does this describe? This described the entire set of integers. Then we look for, okay, if every integer that's not odd is even. This is actually expressed right here as a conditional that if you're not odd, then you are even. So this could be rewritten, but we're saying for this conditional for all integers, if you're not odd, then you're even. And be aware that even and odd are not defined to be antonyms of each other. So if you're an odd, that would be something like n equals 2k plus one. If you're not odd, that just means you cannot be written as 2k plus one. To be even, that means you can be written as like two times L. And so the fact that you can't write it like this implies that does require proof. But again, we're not proving this right now. We're just trying to understand the statement. We have this universal qualifier, universal quantifier, excuse me, attached to the elements described in this conditional. This conditional right here, n is this open variable here. The quantifier then tells you what's the scope of the open statement there. Now let's look at the next one. There is an integer that is not even. This is also a true statement, but notice the quantifier is different now. There exists. This would be the existential quantifier. There exists an integer. We would describe that in the following way. You have the backwards e, which means there exists. And then the same thing before, n is inside the set of integers. So these statements say the same thing. n inside the set of integers means, we're talking about an integer right here. And so you have for every integer or for some integers, okay? And then the rest of it is straightforward. There is an integer that is not even. So there exists an integer such that it's not even. That word not here does go into the symbols. Now be aware that from the previous statement, if you're not odd, you're even, you can also buy a similar proof show that if you're not even, you're odd. But that's not what this statement says, even if we could infer it from it. This statement in symbolic form becomes there exists an integer that is not even. Look at a third example here. For every real number x, there is a real number y for which y cube equals x. This is a true statement. This one actually has two quantifiers in it. And in fact, we have different quantifiers. The first one says for every real number x, we would write this as for all real numbers x right there for all x inside the reels. Then the comma gives us the next quantifier. There exists a real number y. We have the existential quantifier there. So we would write this as that. There exists a y inside the real numbers for which then the equation holds like so. The equation is a statement which is already in symbolic form. Nothing has to be said about that. And then a fourth example, given any two rational numbers a and b, the product a, b is likewise rational here. Now notice in this case, we have two quantifiers again, given any two rationals a and b. So given any, this suggests it doesn't matter which ones you choose. So we could write this as for all a and b inside the set of rational numbers. That's the same thing as saying given any two rational numbers a and b. Now, when this situation, there's no preference placed on a and b. I mean, we wrote them in alphabetical order, but if you switch the symbols, it wouldn't make much of a difference at all. We're gonna see in a second that for these ones over here, if you switch the order of the quantifiers, that actually can make this statement false. But that's because here x and y are not interchangeable. In this case, you cube y to get x. If you cube x to get y, that could very much change things. Now in this one, because multiplication is commutative, a, b is equal to b, a, it doesn't matter who's on first with regard to this. And as such, we don't list two statements as like, oh, for all a inside of the q and for all b inside of q, that would be proper. But because it doesn't matter which one we consider first, we can abbreviate that using this symbol right here for all a comma b inside the set of rational numbers. Now let's consider some false statements. That is, these are statements which are false. I'll let you try to prove these things, but do involve quantifiers. Every integer is even. This is a false statement. Every integer like we considered before for all n inside the set of integers is even. That's our statement. Clearly this is false. I can give you a counter example, take n equals seven. That's not an even integer, okay? It's a false statement. Here's another one here. There exists an integer n. That would mean there exists some n inside the set of integers for which n squared equals two, okay? Again, this is a false statement that there is no integer who is the square root of two. Excuse, yeah, there's no integer whose square is two. That is the square root of two is not an integer. We'll actually prove later on in this lecture series the square root of two is irrational, but we'll do that at some other time. And I want you to be aware that when you look at the quantifiers, the truthfulness does depend on things. Like we looked at this one earlier, all integers are even, right? For all n inside of z, n is even. That's a false statement. But if we were to change that to there exists an n inside of z, that n is even. Well, yeah, sure, you could take n as two. That's an example. So the quantifier can change the truth value. So you have to be careful of that. Now in this example right here, we have an existential one. There exists an integer whose square is two? Nope. If you swap the quantifier for all integers and squares two, that's still false, right? So it does matter here. And so this is one principle you can have that if you have a false statement for the universal quantifier, it might be true for the existential one, right? Just because every number doesn't have the property, it could be that some do have the property. Now conversely though, if you have a false statement involving this existential quantifier, if you swap it to universal, it's still gonna be wrong because if there's no number with the property then it's still false that every number has that property. Now look at example C right here. This one looks very familiar to the one we just considered a moment ago. For every real number X, there exists a real number Y for which Y squared equals X. This looks very similar, of course. I changed from Y cubed to Y squared. If we were to write this into the symbolic form, it looks very similar to what we did before. For all real numbers X, you can write that as for all X and R. There exists a number Y. We can write that as there exists a Y inside of R such as Y squared equals X. This of course is now a false statement. If you take like X equals negative one, then there does not exist a real number whose square is equal to negative one, okay? And then let's look at one similar to the last one we just saw on the previous slide. Given any two rational numbers A and B, the number, the square root of AB is rational. This is a false statement, but we can write it similarly like we did before. Given any two rational numbers A and B, the square root of AB is inside the set of rationales. That's a false statement, but like I mentioned before, if you swap it to an existential, then there are situations where the square root of a product of rational numbers could be rational. Like if you take the square root of two times the square root of two, that's the square root of four, which itself too, which is a rational number. So it's false with the universal quantifier, but it is true for the existential. So one has to then pay attention to your quantifiers. The statement could be true with one, but not the other. It could be false with one versus the other. Now, like I alluded to earlier, it's important that we address this concern about ordering of quantifiers. So let's look at this statement C from the previous slide. Not the, well, two slides ago, I guess I should say. When we were looking at true statements, for every real number X, there exists a real number Y such that Y cubed equals X. We already mentioned how this is a true statement. Now, if you were to move around the quantifiers here, okay, that is if you put the existential statement before the universal statement, this actually makes it a false statement. What would it look like? Well, if you put the Y before the X, you would end up with there's a real number Y such that for every real number X, you have Y cubed equals X. So this statement would then be there exists a Y such that for all X, you get Y cubed equals X. But when you swap the order of the quantifiers, this then becomes a false statement. That, let's digest this a little bit. There is a real number such that you give me any real number, the cube of the first number equals that number. So, all right, you're telling me there's a number Y such that it's cube equals zero and one and pi and the square root of two. It can't be all of those things. This is a false statement. And it comes down to the ordering of the quantifiers here. And you might think I'm tricking you by using different symbols, but if I put X here and Y here and you get X and Y, now sure you gotta change the position over here. It's not the symbols X and Y that's in place, the quantifiers that got moved around. The existential, then the universal is false, but the universal, then the existential is true. And so you have to be very, very cautious on the order in which you read quantifiers to make the statement true. And this is of course what we always do with quantifiers. Statements are true or false based upon how they're quantified. So you wanna be used to these two quantifiers in our logic and mathematics.