 In lecture eight, we introduced the notion of a truth table. And in lecture nine here, we're gonna use truth tables to help us determine when two logical statements are logically equivalent. Well, what does it mean for two statements to be logically equivalent? We say that two statements P and Q are logically equivalent if they have the same truth value for every truth assignment of their variables. And we actually denote this with equality. We say the two statements are the same thing. This would be the same thing as saying like X plus Y squared is equal to X squared plus two, two X, Y plus Y squared. In the algebraic sense, these two quantities are equal. They're not the same expression because I actually wrote two different things, right? But we mean that on every assignment of the variables, they're always gonna be equal to each other. And so logically equivalent statements, we really think of as equal statements. Now to show that two statements are logically equivalent, what you can do is you can construct a truth table that involves the statement P, that involves the statement Q, and show that their columns and the truth table are identical. Unlike the algebraic equations we were considering with statements, there are only a finite number of assignments you can have. It's gonna be a power of two based upon how many primitives are in the compound statements. And so if you consider every possibility, you can then exhaust all the possibilities. And show that they are equal to each other. So what I'm gonna do is I'm gonna first show that the statement not P or Q is logically equivalent to the statement P implies Q. And I'm gonna do that with a truth table just right now. So we're gonna take a second to draw such a truth table, right? There's only two primitives involved here, P and Q. So I'm gonna start off with my two columns, that key track of their possibilities. So we have our primitives P and Q. Then we think of the compound statements that involve P and Q. There's a not P that's gonna come into play. And then for reasons that aren't apparent yet, I'm also gonna put a not Q there. So then we can do like a not P or Q, and then we can do P implies Q like so. We then draw, we then are gonna separate our columns with a line, just to make it easier to read, like so. And then because we have two primitives, there are four rows that we're gonna put for the four possible truth values we can have. And so remember the possibilities are gonna be true, true, true false, false true, and excuse me, false false. Looks like I put an extra row in there. Whoops a daisy, not a big deal. So then for not P, not Q, we just switch the signs, false false, false true, true false, and then true true. Then we get to the compound statements we're interested in, not P and Q. So if you're not P or Q, remember not is gonna be true if any part of it's true. So we end up with a true right here, a false, a true and a true. And then if you look at P implies Q right here, remember a conditional is only false when the premise is true, but the conclusion is false, which happens in this scenario right here. And then on the other ones, here you get true implies true, that's true, and these ones are gonna be vacuously true. So you end up with true and true. And so then when you compare these columns with each other, you then can see that they're the exact same columns, true, false, true, true. So for every assignment of the primitives, these things agree with each other. So these statements are actually the same logical statement. A conditional is just the same thing as saying not P or Q, which I want you to think about this, a true, a not statement is, excuse me, or statement is true if any piece is true. We had mentioned before that if the premise is not true, that makes the statement vacuously true. So it's gonna be true in that situation. But we also mentioned that if the conclusion is true, that will make the conditional trivially true, like we've discussed previously. And so with that discussion in mind, it then becomes self-apparent that, yeah, this is a different way of describing a conditional. These two statements are logically equivalent. What I wanna do next then is show that the statements not P and not Q are not the same thing as not P and Q. That is, those give us different statements there. If I take not P and Q versus not P and not Q, let me draw a line there to make it easier to read. And then let's fill in the truth values here. So not P and Q, you look at these ones right here, not P and Q, that would be true, but then you switch the sign to be a false. Then, because and statements are true, both of them true, but if there's any part that's false, they're gonna be false. So what's gonna happen here is you have true and false, which gives you false, but then not is gonna give you true. And then you can fill in the other two as true and true as well. Then with the last one here, not P and not Q. So we're gonna look at these ones right here. False and false is false. You have here, false and true, that is a false statement. You have true and false, that is a false statement. And then you have true and true, which is a true statement right there. So now when you compare these ones side by side, sure, false and false, true and true, there's agreement there, but the disagreement happens right here. One is true and one is false. And so because they do disagree with each other on some of their truth assignments, this shows that the two statements are not logically equivalent. They're not the same statement. Taking not of an and statement is not the same thing as taking and of not of negated statements there. The order of operations makes a big difference there. Let's do two more examples of showing that statements are logically equivalent here. On this example, we're gonna construct a table to show that the bi-conditional statement P is equivalent to Q is the same thing as saying P and Q or not P and not Q. Now, our table is not gonna need to have a lot of rows because we only are gonna have two primitives in play here which exactly are gonna be P and Q, mind you. Let's see, what are the things we'd have to consider? This one we could consider now but I actually wanna put it at the end. So some things I wanna consider, let's have a not P, let's have a not Q, let's have a P and Q, let's have a not P and not Q. Then let's put them all together, P and Q or not P and not Q. And then at the very end, we're gonna have P is equivalent, can't even read that, P is equivalent to Q right there. Whoops, Q, line our table. Again, you don't have to make it look perfect but hopefully you're trying to make something that looks organized, like so. And so then we need four rows. So draw three more lines and that one's a little bit slant. I can live with it, not a big deal. Oh, that one's slant too. Again, we're just gonna move on. Don't have to be too nitpicky about this one. So what are your four possibilities? True, true, true false, false true and false false. Oh, I put an extra line in there again. Whoops a daisy. Negation switches things, false false. False true, true false and true true. All right, so now we have to do an and statement. This would be true, false, false, false, not P and not Q. So we're looking at this one this time. So we get false, false, false true. Now this here is an or statement. So looking at these two columns, do or with that. If any part is true, the whole thing is true, true, false, false and then true. And then for the bi-conditional, remember how a bi-conditional works? A bi-conditional is true if they have the same true value and it's false if they have different true values. So looking at these two columns here, true and true makes it true. True and false is false. False and true, that would give you false again. And then lastly, false and false, that's actually a true value there because they have the same true value here. So then looking at these two columns, we see that they're the exact same thing, true, false, false, true. And so that then proves that these statements are logically equivalent to each other. But that kind of makes sense when you look at it. It's like, okay, if either both are true or both false, if one of those happens, that makes this a true statement. That's exactly what bi-conditionals do. And like I said, we're gonna do one more example here. Let us make a table to show that P implies Q is the same thing as not Q implies not P. So what do we wanna consider here? Well, we need to have a P, we need to have a Q, a not P, a not Q. We should have a P implies Q and we should have a not Q implies not P. Oops, I want that to be yellow. I want that to be straight. There we go. I'm of course doing this by hand so that you yourself can mimic this if you're trying to write these in your notes as we do this together. So my four possibilities are true, true, false, false, true and false, false. If you negate those, you're gonna get false, false, false, true, true, false and true, true. For a conditional looking at the first ones, remember it's only false when the premise is true and the conclusion is false. Otherwise, they are true. That's how conditionals work. So now looking at this one right here, we need to look at these columns and remember a conditional is only false when the premise is true, but the conclusion is false. So you're gonna get a false right here, but in every other situation, these statements are vacuously true and this one right here is triply true. And so then when you compare these columns together, you see you get the exact same thing. So these statements are in fact logically equivalent. Now the first one, let me note here, this is the conditional statement P implies Q. This right here, not Q implies not P. This is the contrapositive, the contrapositive, which we introduced previously and this actually gives motive on why we care about the contrapositive. The contrapositive is logically equivalent to the original conditional statement, which means if this one is true, so is this one. If this one is false, so is this one. And sometimes when one works with a proof, you wanna prove a conditional statement, believe it or not, sometimes working with the contrapositive is easier to prove. Therefore, if this is proven to be true, that means this is true as well because they're logically equivalent to each other. Now, I'll leave it as an exercise to the viewer here to show that the conditional P implies Q is not equivalent to its converse, Q implies P. So the converse and the conditional are in fact different statements, okay? But what you can show also, I'm leaving this up to the viewer here, you can show that the inverse, not P implies not Q, is logically equivalent to the converse. And this is actually not too hard to see because the inverse is actually the contrapositive of the converse. And so the fact that contrapositives are equivalent, known as the law of contraposition, because of that, these two will be logically equivalent. But like I said, I'll leave it up to the viewer to show that a statement and a converse are not logically equivalent to each other.