 Hello and welcome to this video on truth tables for conditional statements. So we talked about the concept of a truth table at the very end of the last video. We were looking at conditional statements and determined that a conditional statement that is something of the form if p then q is true in every circumstance except one. That is when p is true but q is false. This is a promise that isn't delivered upon like saying if you eat your dinner then you can play outside and you do eat your dinner but then you're not allowed to play outside. It's a lie. It's horrible. That's the situation where the conditional statement is false. And if p is false, the hypothesis is false, then the entire promise is actually said to be true because the statement said nothing about what should happen when the hypothesis isn't met. If you don't eat your dinner then your parents are free to do what they want. But notice something really important here and that is the actual contents of the statements p and q don't matter. If I had another conditional statement like the one in the concept check, if it's cold outside I'll put on my gloves. And this time p is the statement, it's cold outside and q is I will put on my gloves. The statement is still false under the same conditions, namely when p is true but q is false. And the statement is true in all situations otherwise. So it's the form of the statement rather than its content that determines whether the statement is true or false. This is really important for understanding how to think logically. If you're in a political debate for instance with somebody you might be able to point out flaws in an argument without even understanding all the terms in the argument because the person made an if then statement that had a true hypothesis but a false conclusion. So let's leverage this fact to introduce an important little gadget for understanding complex statements called the truth table. Let's go back to the statement, if you eat your dinner then you can play outside. And let's set up a table that records whether or not the entire statement is true for each possible situation that could happen here. So I'm making a table with three columns. One column will record whether the hypothesis condition is met or not. The middle column records whether the conclusion happens or not. And then the last column will record whether the entire statement is true or false. Well if the hypothesis is true and the conclusion is true we've seen that the entire statement is true. So I'm going to put t, t and t in a row here for true, true and true. Another situation is when the hypothesis is met like the kids eat dinner but the conclusion does not happen. The kids are not allowed to play outside. So I put a t here in the hypothesis column and an f here in the conclusion column because the hypothesis was met but the conclusion didn't follow. And in this situation we've seen that the conditional statement is false. So I'm going to put an f in the final column. A third situation is when the hypothesis is not met but the conclusion does follow. And a fourth situation and in this case the final situation is when the hypothesis is not met but the conclusion and the conclusion does not follow. Let me start those two rows together false, true and then false, false. Now here's a concept check. What goes in the last two cells of this table? Is it true then true? True then false, false then true or false then false? The answer here is A, both true. Remember this is the counterintuitive part of conditional statements. This is because if the hypothesis is not met, remember I'm under no constraint of my behavior based on the original statement. That statement does not say what I should do if my kids don't eat their dinner, I'm free to do what I want. So regardless of what I do, let them play outside or not, my original promise is still true. So here we have a completed truth table that it shows exactly when the statement is true and when it's false. And since it's only the form that determines the truth and falsehood and not the actual content of the statements, we've really created a once for all time rubric for knowing when a conditional statement is true or false. Let's fix some notation and say that the statement if P then Q is notated by this P with a right pointing arrow to Q. This notation makes sense because in a conditional statement when the hypothesis is satisfied then it should naturally lead to the conclusion, that's the arrow. So we sometimes say P implies Q. It's just another way of saying if P then Q. So for all conditional statements P implies Q, we have this truth table. So in other words, P implies Q is true in all cases except one. P is true, but Q is false. Now you should memorize this truth table or better yet, just understand the logic and the language behind why this conditional statement is true when it is and false when it is and be able to reconstruct it on the spot when you need. We will need the results in a major way as we move on to thinking about how to prove that a conditional statement that we believe to be true really is true. See you then and thanks for watching.