 Previously, we've talked about when an implication can be vacuously true. I wanna continue that discussion in this video but also talk about something very important that's related to as well. When can an implication be trivially true, okay? Now, to understand what this terminology means, let me remind us the four possible evaluations we can have with a conditional statement. A conditional statement has a premise P and a conclusion Q. These are primitive statements that themselves can be true or false. The true values of the premise and the conclusion then relate to the truth value of the conditional which is itself a compound statement. But they aren't, they're not necessarily the same but they are related. And these are the four possibilities. Your premise can be true and your conclusion could be true. That makes the conditional true statement. It could be that your premise is true and your conclusion is false. That actually makes it a false conditional statement. This is the only time a conditional can be false because you have some expectation. If this happens, then this will happen. But it happened, but not this one. That's a lie, that's a false statement. If the premise is false and the conclusion is true, that makes it a true conditional. Also, if the premise is false but the conclusion is false, that also makes it a true conditional. This second case we've talked about before and we'll talk about it again, of course, in this video here. If the premise is false, then the conclusion is under no obligation to be true or false. And therefore the conditional statement is still a true statement, all right? So let's talk about how does the truth value of the premise or conclusion affect the truth value of the conditional. Now, look at these two cases right here. The case where true implies true and false implies true. In both situations, the conclusion is true and the conditional is true. And so this is an important thing to realize. If the conclusion of a conditional is true, then the conditional is likewise true. If the conclusion is true, then the conditional is true. And so we can use that as a proof pattern to show that a conditional statement is true. If the conclusion is true, independent of what the premise is, it doesn't matter if the premise is true or the premise is false, it don't matter because if the conclusion is true, that makes the conditional true. And so we can argue that a conditional is trivially true because its conclusion is true. Let me show you some examples of these things. These are very, very simple propositions that you are not gonna win the millennial prize with this or anything like that because they are trivially true. Take any real number x. If here the square of x is less than 73, then zero is less than one, okay? Now, depending on your choice of x, x squared may be less than 73, maybe not. But regardless of this, when you look at the conclusion, zero is less than one. The conclusion zero is less than one is always true. And therefore the implication is true. Let me be clear about this. If P is the statement, x squared is less than 73. And if Q is the statement, zero is less than one, then with regard to Q, we know Q is true. With regard to P, we don't know. It could be true or false depending on the choice of x. But the implication that P implies Q, that is a true statement regardless of whether x squared is less than 73 or not. Because if it is, then this will be true. But even if it's not, this will still be true. And this is why of course we call it trivially true. It didn't actually depend on the premise. Now, typically when one writes a mathematical proof, if a statement doesn't require the premise, one typically just ignores the premise because it's not doing anything. You're sort of overly complicating it. But from a logical sense, this is a valid proof. Let's look at another one. If A is an integer such that A is odd, then two A is even. Well, I want to point out again that the conclusion is always true regardless of the premise there. Because even means that you're divisible by two. And divisible by two means you can factor the numbers two times A. The number two times A is clearly divisible by two. And hence the implication will again be trivially true. This, since the conclusion is always true, the implication P implies Q is also true. I'm not saying that A is odd. A may be or may not be. But the implication that P implies Q is true because the conclusion Q is true. Now, when we talk about this being trivially true, I want you to note that the triviality of the proof is because the conclusion is true, then the conditional is true. It doesn't mean that it's trivial, that the conclusion is true. We're saying that the conditional is trivial, but the conclusion actually could be elaborate, right? And that oftentimes happens, right? We need something to be true. We have some hypotheses. And then we end up not using the hypotheses because the conclusion was true for other reasons, but it maybe wasn't self-apparent. So consider this one here, this proposition. Let X be a real number. If X is less than five, then X squared minus two X is greater than equal to negative one. Now, be aware that this inequality, the conclusion we're trying to prove here, X squared minus two X is greater than equal to negative one. That is algebraically equivalent to the inequality X squared minus two X plus one is greater than equal to zero. You get that just by adding one to both sides of the equation, okay? And then on the left-hand side, if you take X squared minus two X plus one, you could factor that. That turns out to be a perfect square trinomial. It becomes X minus one squared, which is greater than equivalent to this one right here. So these two inequalities are equivalent. These two are equivalent. That means they have the exact same solution set. But then this one over here, it turns that if you take X to equal one, you'll get zero. So you have equality. And then if you take something other than one, you're always gonna get a positive number. This is properties of real numbers. Think about geometrically, you're just taking the graph of some parabola, greater than or equal to zero. You're talking about the Y coordinate there. It's always above the X axis. This is always true. And it doesn't matter whether X is less than five or not. This holds always. And therefore, this is a true statement because the conclusion is true. And the reason I present this example is that when you first looked at the conclusion, it might not be obvious from how we wrote it that that's a true statement. Like in this case, it was obvious. But the inequality, maybe you didn't see it at first. If you did, good at you, but you probably didn't see it at first. But one can then show that the conclusion is true. And therefore the implication, the if then statement is also true. So if the conclusion is true, then this implication is true regardless of the premise. Now, in situations like this, if you prove something is true without a hypothesis, that is a valid proof. But when it comes to mathematical writing, if you didn't use a hypothesis, that doesn't make your proof invalid. But what you typically do is you remove the hypothesis. And so you say something like this, let X be a real number, and then X squared minus two X is greater than or equal to negative one. You can strengthen the statement by removing hypotheses that are unnecessary. You don't have to remove the hypothesis because the statement would still be true even with superfluous hypotheses. But better mathematical writing happens by removing unnecessary hypotheses, even if logically it's not necessary. Now, on the flip side of this coin, I want us to consider the cases, we've considered this before, consider the cases where now the hypothesis is always false. So we saw that if the conclusion is always true, the implication is true, but if the hypothesis is false, what does that say about the conditional? Well, there's the situation where false implies true and there's the situation where false implies false, that is always a true statement. Like I alluded to earlier, if the hypothesis is false, the conclusion statement has no obligations. It could be true, it could be false. And this is what we mean by something being vacuously true. Vacuous because the premise is empty because it's false, but the conclusion could still be true even if the premise is false or it could be false, right? The conclusion, but the statement, the conditional statement is true is what we're trying to offer here. You can put on any ridiculous premise you want, if it's false, and then your conclusion could be anything. You could ask for the moon at that point because it will be true, vacuously speaking. Let me give you two examples of this. Let X be an integer. If X is greater than three and less than two then X squared plus four equals seven. I want you to notice that this statement, this if then statement is a true statement here. Let's be careful here. You have P is the statement three is less than X and X is greater than two, excuse me, X is less than two, like so. And then Q is the statement X squared plus four equals seven. These are our statements here where considering the implication P implies Q. Is that a true statement? Well, with regard to the conclusion this time, we don't actually know. Depending on your choice of integer X, it could be true, it could not be true. Actually, one could do even better because if we restrict X just to be integers, are there integers whose square plus four is equal to seven? That would be the same thing as X squared is equal to three and there's no integer whose square is equal to three. So in this case, we can actually determine that the conclusion is false. There's no integer whose square plus four is equal to seven but regardless of that, when you look at the premise, if X is greater than three but less than two, then this will follow, right? But the premise is likewise false. There is no integer that's greater than three and less than two, it's not possible. So this is a false implies false situation but like we observed before, the conditional is true. So in common English, this is when we use idioms like, well, when pigs fly, you can go to the concerts or whatever, maybe a parent's telling you that you can't go to the concert that all the cool kids are at or whatever. The idea is they give you some hypothesis that, oh, okay, if this impossible thing happens, then you can go, right? So their conditional is true. It's being a little mean of course but it is a true conditional statement that I don't care if the conclusion is false because the premise is false, the implication is true. This being true does imply that being true even if both statements are impossible, the conditional is true. Another example of this, let X be again an integer and suppose that negative X squared is greater than two, then X equals five, right? Well, in this case, the conclusion could be true if X is five or it could be false if it's any other integer, right? But when you look at the hypothesis, negative X squared is greater than two. When you take a real number like an integer and you square it, that always is going to give you something non-negative. It'll usually be positive unless it's zero, in this case, it could be zero. So then when you put a negative in front of it, all those positive will turn to negative and zero will stay zero. None of those are greater than two. So in this situation, the hypothesis is false and as such, the implication P implies Q here is a true statement. It's true because it's vacuously true and that is the proof. Now again, people typically don't worry about proving vacuously true statements or at least you might not think it. It turns out they do show up with enough occasion that's worth consideration here. But what I'm trying to say here is that when we work with conditionals, there are some special cases where it's true trivially because the conclusion is automatically true. I should say the conclusion is true therefore the conditional is automatically true. There's also this case where the conditional is vacuously true because the premise is false and therefore the conditional is true. Now because of these considerations, one must determine the true value of a conditional statement such as P implies Q. There's basically only one case you really care about. It's the case where false, excuse me, where true implies false, which we know that of course is equal to false. That's the one we care about. And so you have to determine that never happens. That true implies true, that's fine. False implies true or false implies false, those are fine. You have to show when you're working with an implication that this never happens. And so the way you do that is you assume you assume P is true because that's the only situation this could happen. If P is false, you're already done because it's vacuously true. Then you argue, argue, argue, logically speaking and then you conclude, oh, therefore, therefore Q is likewise true, thus avoiding this situation right here. This is the basic template of what we refer to as the method of direct proof. One proves an implication by assuming the hypothesis and then arguing the conclusion is true under that hypothesis. Now, of course, you can have trivially true statements where the conclusion is true regardless of the hypothesis but the direct proof only worries about this one case because the other three cases are trivial or vacuous in those situations. We just have to make sure we avoid this situation. And that brings us to the end of lecture seven. Thanks for watching. If you learned anything in this lecture, please like these videos, subscribe to the channel to learn more about these things, to see more math videos like this in the future. And of course, post any comments below if you have any questions and I'd be glad to answer them.