 Hello there! In this quick screencast we're going to do three examples of where we're going to use truth tables to establish whether two propositions are logically equivalent or not. Logically equivalent statements have the same truth values and under all the same conditions, and so if we order the rows of their truth tables the same we're going to see the final columns always come out the same. Let's start by looking at this example here where we're going to show that not p or q is logically equivalent. That's what that symbol means to not p and not q. This is the second part of what's known as De Morgan's Laws, one of many laws of logical equivalence. De Morgan's Law, really part two. So the way we can use truth tables to decide whether the left side is logically equivalent to the right is just to make a truth table for each one and see if it works out the same. Both of these statements involve only two small statements, so we're going to need a column for p and a column for q, and then just four rows. We're going to alternate true, true, true, false, false, true, false, false, and we're just going to make a truth table for the statement on the left hand side, a truth table for the proposition on the right hand side, and then see if they give us the same results in all the same conditions. So let's work with the left hand side. I'll use a purple color for that one, so this side here. Building this truth table up from the very basics, we're going to need a column for the disjunction in the middle, that's p or q. So let's make that really quick. This is a disjunction, and so we're going to have a true statement unless p and q are both false. If either one or both of those individual statements is true, then the disjunction is said to be true. So this is going to be true here, true here, true here, false here. And now I'm going to need a new column for the negation of that statement, so not p and q, or p or q, sorry, just take p or q and change the truth value. So false here, that's a false, false here, false here, true here. Okay, so that is the first statement that's involved here. I'm going to switch colors over to red and do the right-hand side here. Now I'm just going to do this in the same basic area, not make a brand new truth table, just use the existing p and q columns to make this new one here. So I'm going to need, let me make a dividing bar here though, just kind of split this off, separate it from the others. So let me make a column for not p and a column for not q, because those are two sort of smaller statements that make up the proposition on the right. Now not p, I'm going to get by going all the way over to the column for p and changing the truth value. So false, false, true, true. And the same thing for q, just flip the truth values, false, true, false, true. And now I will make a final column for not p and not q, the conjunction. And then I'm just, to do that, I'm just going to look at these two columns here and join them with an and. And so this is going to be false unless both this column and this column read true. False and false, I guess we false, false and true, I guess we false, false here, and then true down here. And just to wrap this up here, to notice, let me just get the highlighter here. And you notice that in this column, I have false, false, false, true. And this column over here, for the other statement, I have false, false, false, true. So those two statements are logically equivalent because their truth tables give the same results under all the same circumstances. Now let's look at another example of logical equivalence where we're going to show that p implies q, basic conditional statement is logically equivalent to not p or q. This is a handy way to rewrite an implication if you ever needed to do so. I've already set up the truth table here. There are only two variables, so four rows. I'll need one column for this right hand or sorry, this left hand statement p implies q. And this is just a straight implication. And so this is going to be a true statement unless the hypothesis p is true and the conclusion q is false. So we have true, false, true, true. Let me change color here to red. And let's take a look at the right hand side here. Let me just make a bar to separate it off. I'm going to build this up piece by piece. And so one piece would be not p. And again, that's just going to be obtained by flipping the truth values of p. So false, false, true, true. And finally, I'm going to need a column for the ending statement, not p or q. I'm going to look at not p right here and q over here. And if this is an or statement, so if either one of those two statements here or here is true, I have a true disjunction. So in the first row, I have true, false. So that gives me true. In the second row, I have false, false. So that gives me false, true, true. That gives me true, false, true. That gives me true. And once again, if you look at the results here, for this proposition, and this proposition have the same truth values in all the same circumstances, therefore they are logically equivalent. Okay, so finally, let's show that a couple of statements are not logically equivalent using truth tables. We're going to work with a pair of propositions that showed up earlier in the semester for us, not p, parenthesis, and q on the one hand, and then not parenthesis, p, and q on the other hand. So a question might arise is do those parenthesis really matter? Does it make a difference? Do we need to put those in there? Well, in the sense that it matters, we want to see if those two statements are logically equivalent. If they are, then the parenthesis don't matter. If they are not, then they do matter. So let's build up these one by one. So let's underline these blue for this first statement. I'll make a column for not p, and that would be false, false, true, and then just one more column for not p, parenthesis, and q. So this is an and and so I'm looking here and here for my component statements, and both of those statements need to be true in order for this entire statement to be true. So I'd have false here, false here, true here, false here. And now let me just make a dividing bar, switch over to a different color, and let's look at this one. So I'll need to make a p and q column first. I'm going to go to here and here for that. This is just a straight conjunction here. So I'm going to have true, false, false, false, and then I need to negate that. And this will be my last column. So not p and q. I'm going to get that by going here and flipping the truth value. So I have false, true, true, true. Now let's take a look at the results here. And you see that in a couple of cases, so here and here, and this last row, these two statements here have different truth values under the same conditions. And so that makes them not logically equivalent. Okay, so that's three examples of how to use truth tables on simple two variable statements to prove logical equivalence of statements or to show that two statements are not logically equivalent. If you move up to three variables, it's the same principle. And if you have issues making a three variable truth table, please see the video about how to do that. Thanks a lot.