 So welcome to the screencast on quantified statements. So this is a concept that goes back to an earlier concept we saw in the last section of an open sentence. There's a definition we saw, basically an open sentence is a sentence, it's either an English statement or a math expression or a hybrid of those two things. That's almost a statement in the logical sense that has a definite truth value, except an open sentence has a variable involved with it. And so we can't tell the truth value of that sentence without knowing a little bit more about the variable. Here are three examples of open sentences. Each of these, in each of these cases, there are variables present. Whether it's k or x or even two variables, x and y. And each of these sentences would have a truth value if we knew something about the variable. For example, to say the integer k is even, is that statement true or false? Well, we don't know, because we don't know what k is. Is the statement x plus y equal to false or true? We don't know. I mean, for this statement, the sentence is sometimes true for certain values of x and y, like one and one or five and negative three, but not for all values necessarily. And even the sentence x squared bigger than or equal to zero, is that true or false? Well, we can't really say until we say more about what the x is. Is x a real number? Is x an imaginary number? Is it an integer? We don't know. We can't tell the truth value of these sentences without knowing a bit more about the variables. Now, if we assume just for the moment that the universal set is the set of all real numbers. We can say something about this sentence right here. If x belongs to the real numbers, then we somehow know that through our basic arithmetic that this statement is really a statement. And it's actually true for every real number x. So we can say something about how often this sentence is true. And we can say something about how often these two sentences are true. And that's kind of the gist of the screencast. Given an open sentence, how often is that sentence true? So we're going to open this up with a concept check, actually. Let's lock down the universal set to be the set of real numbers. And look at the open sentence x squared minus one equals zero. Now, how often is that true? Is it always true, sometimes true, or never true? And the answer here is, of course, going to be sometimes, but not always, right? Sometimes, because there do exist x values that make this sentence true. For example, x equals one makes this statement true, and x equals minus one makes this statement true, but not always, right? We're pretty well aware that, for example, if x equals two, yeah, the sentence does not really, it's a statement, but it's false, okay? Two squared minus one is not equal to zero. Now let's do another little concept check here. Same kind of question here, but different open sentence. So look at the open sentence, x squared plus one is bigger than zero. And again, the universal set, we'll just consider x to belong to the real numbers. Now, is this sentence true always, sometimes, or never? And in this case, the answer is always. Okay, when you take a real number, okay, the universal set really matters here. We take a real number, x squared by itself is bigger than or equal to zero. If we add one to it, that makes that inequality strict. So this is a statement that is always true. It is true for all real numbers x. So in both of these concept checks, we introduced open sentences and asked how often is that open sentence true? And sometimes the question is sometimes, sometimes the question is always. And in future screencasts, we're going to take up the situation of what happens if a statement is never true, what can we say about that? So let's pull out the main concept here. Take a look at the first open sentence from the concept check that said x squared minus one equals zero. Now that is not a statement as is. We don't know enough about the variable to say whether this is true or false. We can't determine the truth because we don't know for which values of x this is supposed to be true or false. And for all of them, for some of them we don't know. We don't even know what x is. However, if we add this little bit of verbiage onto the front end of that sentence for some x in the real numbers, remember the symbol right here means is an element of. So for some x in the set of real numbers, x squared minus one equals zero, then that is a statement. In fact, it's a true statement. What we've done here is we have quantified the variable. I have said how often should this open sentence be true? We said for some x in the real numbers, sometimes at least one x in the real numbers, this is a true statement. We've quantified the variable. And so we've turned an open sentence into a legitimate logical statement whose truth value we can apprehend. In shorthand, we're going to write that phrase for some x and r with a special symbol, this little backwards e right here, which you should read. There exists. There exists. Okay, so the way you would read this often in English is to say there exists an x in the real numbers such that x squared minus one equals zero. And that happens to be a statement, happens to be a true statement. The way that we've quantified this is to say that there exists an x in the real numbers that makes this equation work. And so we're going to call this an existential quantifier. Existential quantifier, okay? That's what this is. We're saying that given this open sentence who's, it's not really a statement because we can't tell the truth value. If I quantify this existentially by saying there exists an x, at least one x, possibly millions of them. But at least one x in the real numbers that makes this equation true. What we've done is we have quantified the variable and turned an open sentence into a statement whose truth value we can determine. Now on the other hand, take a look at the set, the open sentence from the second concept check, x squared plus one greater than zero. And again, the universal set we've fixed down to be the real numbers. So that's not a statement either, although we feel like it ought to be true. We still need to say something about that variable. How often should this be true? If I add this verbiage onto the front end of that open sentence and say for all x in the real numbers, not just there exists an x in the real numbers, but for all x in the real numbers, x squared plus one is bigger than zero, that's a statement for sure. And we, again, we've quantified the variable. We quantified it in a slightly different way. Well, maybe not slightly different way, possibly radically different way. What we've done is say, this open sentence is true always. Okay, and that's much stronger than saying that it's true sometimes. So we've quantified this variable, and when we say for all x in the real numbers, we quantify an open statement by saying the statement is true always. Then we are going to use a special symbol for it too. Kind of an upside down a, which you should read for all. Okay, so the way this shorthand would read in English is to say for all x or for every x in the real numbers, x squared plus one is bigger than zero. So once again, we've quantified the variable in an open sentence, thereby making it a statement. That statement happens to be true this time. And since we have quantified this for all x, we're going to call that a universal quantifier. So an existential quantifier is where we slap a quantification onto an open sentence to say that the open sentence is true for some values of x. At least one value of x. A universal quantifier is where we do the same thing. We specify how often the sentence is true, but we specify that it is true for all x that belong to the universal set. So let's do a couple more examples, four more examples actually. Here are four quantified statements. If you just looked at this second set of parentheses, those would all be open sentences to say that square root of x belongs to the integers. We don't know, we have to say something about the variable. And so that's what the first set of parentheses is for. Those are all quantifications, let me get rid of those two things. So which of these is true? Okay, so is it true that there exists an x in the integers now, such that the square root of x is also an integer? Well, that is certainly true. And how do I know it's true? Well, all I'm saying is that there exists an x whose square root is an integer. So I could pick one out. I could say, what about x equals nine? Okay, that's an integer. And at square root three is also an integer. So this quantified statement is true. Now look at the same open sentence with a universal quantifier rather than an existential quantifier. We're gonna say in this one that for all x in the integers, the square root of x is also an integer. Now that is false. How do I know it's false? Well, I'm making a claim that this statement, that radical x is an integer, a whole number, is true for every single x that is also a whole number. And that's a simply false because look, x equals five. Try it, you try the square root of five that is not a whole number. So this statement is not true because I can find a counter example. Moving on to this third one, is it true that there exists an x in the integers such that x cubed is an integer? Well, that's certainly true. If all I'm claiming is that there exists such an x, I can just pick an example out, like x equals 10, okay? x equals 10, 10 cubed is 1000, that's an integer, okay? Fantastic, it's true. But we can actually say more, the fourth statement here says that for every x in the integers, x cubed is an integer. Now that is a strong statement, but it happens to be true. Now I can't just use an example to prove that this statement is true. Because as we've seen before, when I make a general statement about every single element in a set, I can't just pull an element out and expect it to be representative of every single element. I have to do something else. One way you could see that the statement is true is by using the closure properties of x. So if x is an integer, then x times x is also an integer by closure. And so therefore, x times x times x is also an integer, which is x cubed. So that would be a way to explain that no matter what integer you take, its cube is also an integer. So there's the difference between existentially quantified statements and universally quantified statements and just a beginning of a taste on how to determine whether such things are true or false.