 So in the previous video, we introduced the notion of a conditional statement and talked about what a conditional statement is. And in particular, we talked about when is a conditional statement true? Remember, the only time a conditional statement is false is when the hypothesis is true, but the conclusion is false. This is actually a false conditional because you didn't keep the promise in that situation. The guarantee failed that this did not imply the second one right there, okay? Now, there's a little bit one has to pay attention to when the statements involved are themselves open statements. So recall that an open statement is a statement where the truth value depends on some variable or maybe some list of variables. There could be more than one in the statement. So consider some of the following primitives right here. Don't take the whole sentence, but like right here, if you take the primitive, n is an even number. We've looked at this before. Well, n is an even number dependent upon what n is, right? I mean, that statement being true depends on what n is. Like if n is six, then it's true, but if n is five, that's a false statement. Similarly, if you look at the primitive statement, n is a multiple of six, then again, the truthfulness of that open statement depends on how you choose the n. If n is 12, then yeah, it's a multiple of six, but if it's five or four, then n is not a multiple of six. The truthfulness depends on the choice of n, okay? That's something we've seen before, that's not new, but how open statements relate to conditionals does deserve a little bit of attention here. So I've taken those two primitives here. So this primitive, we're gonna call it p, and this primitive, we're gonna call it q. There are two ways we can combine these together, these two open statements to form a conditional. There's the first one, if n is an even number, then n is a multiple of six. So we could write that as p implies q, but then we could also switch them around. If n is a multiple of six, then n is an even number. That would be q implies p using our notion of p and q right here. So when we look at this first conditional here, the truth value seems to shift based upon the value of n right here. So if we pick different even numbers, we see there's some issues here. Like if we take n equals 12, all right? If n is an even number, 12 is an even number, then it's a multiple of six. In this situation, if n equals 12, this is a true statement because 12 is an even number and it's a multiple of six. So that would be great. But conversely, if we take something like n equals four, four is an even number, but it's not a multiple of six. So it would be false in that situation. So this kind of feels like the open statements we saw before that depending upon how you choose n will determine whether this is true or false, right? But then when you look at the second one, you see things a little bit differently here. If we consider the second statement, if I pick, because n has to be a multiple of six, if I pick something like 12 again, well, 12 is six times two, but it's also two times six. That means it's an even number. So that was true. I could pick a different multiple of six, like 18. That's six times three, but that's also the same thing as two times nine. That's an even number, that would be true. And as you search and search for multiples of six, you take things like 102 or 3,144 or let's see 5,961,252. These are all multiples of six, by the way. You can check it if you want to. These numbers are, they're multiples of six, but they're also all even. It's true, right? And search and search as you might, you actually can't find a multiple of six that's not even. In fact, we could actually prove it. And you can see the proof right here on the screen. If n is a multiple of six, then n is an even number. Well, to prove it conditional, what you do is you actually assume the hypothesis and then you have to prove the conclusion from it. So suppose that six divides n. That's what it means to be a multiple of six. Well, if six divides n, that means there exists some integer, k, such that n equals six times k. Again, that's what it means to be divisible as six right there. But by the way, six is factorable as two times three. So you can play with this. n equals six k, but six becomes two times three. So this is the same thing as two times three k. Now, any integer times three is still an integer. So as this right here is an integer, that means n is two times an integer and that means two divides n. And that's exactly what it means to be an even number. Even numbers are those integers that are divisible by two. So we actually can prove this second statement as true regardless of what the choice of n is. We actually can prove the second one that it's always true regardless of n. But this one right here, it was like, it's sometimes true, it's sometimes true based upon what n is, like if n was 12. But it was also sometimes false based upon different choices of n, like in the case of n equals four. And because of these two phenomena, this one was always true and this was sometimes true, we actually have to be more careful when we talk about conditionals when there are variables in play here. All right, so let's talk about this. So with these open statements, by all means, open statements, their truth depends on the variable in place. But when it comes to open conditionals, that is if there's a conditional that has a variable in it, there's actually no such thing as an open conditional. So to explain what I'm talking about right here, let's take two primitives right now that don't have any variables in them whatsoever. So they're closed statements. So take p equals dogs are the best and q equals I should have a pet dog. And so we could look at the implication right here, the conditional p implies q, if dogs are the best, then I should get a dog. This statement, this conditional statement has no variables in it because the primitive statements have no variables in it whatsoever. But then if I come down here and consider the examples we had a moment ago, let's now take as our primitives p of n where n is now the variable in play here, p of n is the primitive statement, n is an even number and q of n is the statement in as a multiple of six. Now the truth value then depends on the variable itself. Like if I take p of six, this is true. If I take p of two, this is likewise true. If I take p of five, this is false. So this open statement is really a function which takes on in this case, the variable is an integer, but the output is a Boolean, true or false. q of n on the other hand, if you take something like p of six, this is true because it's a multiple of six. If you take p of two, this is false. Two is not a multiple of six. And you take p of five, this is likewise false. So the truth value depends on these. We can think of open statements as function, functions, Boolean functions, okay, as in they have a Boolean output. But nonetheless, when we look at the conditionals, the conditional is itself a statement. The conditional as a statement is a statement which could be true or false. When we talk about conditionals, there's no such thing as an open conditional. Conditionals are independent of the variable. So when you look at this conditional, p of n implies q of n, this reads if n is an even number, then n is a multiple of six. We looked at that before. And then there's the second one here, q of n implies p of n in this situation. If n is a multiple of six, then n is an even number. With this one, we considered examples like the following. Take p of 12 implies p of, sorry, q of 12 in that situation. Now be aware, p of 12 would be 12 is an even number. That is a true statement. And then you look at the second one right here, q is a multiple of six. So q of n is a multiple of six. So q of 12 would then be 12 is a multiple of six. That's true. So this would be an example where the conditional is true. So this right here, this instance of n equals 12 is true. And it has the format true implies true, which is equal to true. Now conversely, what if we look at the instance where n equals five? In that situation, you would get something like p of five implies q of five. Well, five is not a even number. So that's false. And then five is not a multiple of six. So that would be false, but the conditional statement itself is actually true. This is an example of when you are vacuously true. So that's a true statement in that situation. The one we have to be cautious about would be something like the following n equals four. In that situation, we look at p of four implies q of four. Now in that instance, this instance where n equals four, you end up with the first one, p of four is true, but q of four is false. And so t implies f is actually a false statement. And so this is the issue we were having before that for some instances, which is why I illustrated in red, just dripping with blood right now, because there are some instances that make the conditional false, that actually makes it a false statement. This conditional with the open values in there is a false statement because there does exist an instance of the hypothesis, which is true, but the conclusion is subsequently false. So the conditional statement is true. If you have something that's like sometimes true and sometimes false, let me make it very clear for you. If you are sometimes true, sometimes false, that actually makes you a false conditional statement. There is no if, there's no if, ands or buts about it. If a conditional statement is sometimes false, that makes it a false conditional statement. If there exists an instance of the variable that makes this conditional false, then it is a false conditional statement. So this is a false conditional statement, even though there are some instances that make it partially true, it's a, the conditional statement is in fact false. This is in contrast to the other direction, Q of n implies P of n. It doesn't matter for any, for any instance n equals k, we have that Q of k implies P of k will be the following. If this is true, this will imply truthfulness over here, that whatever n is a multiple of six, then n will be an even number. This is what we saw earlier with the proof. So the important thing to understand with conditionals is that if the hypothesis is sometimes false, excuse me, if the conditional is sometimes false, then that makes the actual conditional false. There is none of this sometimes true, sometimes false. A conditional is either true or false. If it's not always true, the conditional, that makes it a false conditional statement. And so that'll bring us to the end of lecture number six. Thanks for watching. If you learned anything in these lectures, please like them, subscribe to the channel to see more videos like this in the future. And as always, if you have any questions, feel free to post them in the comments below and I'll be glad to answer them as soon as I can.