 Hello and welcome to the screencast on logical equivalence. So what does logical equivalence mean? It's a very important concept for us in Math through 10. Two statements x and y are said to be logically equivalent if they have the same truth value for all combinations of truth values for all the variables in the statements. Now what that means is if we made a truth table for x and a truth table for y and both x and y could be made up of smaller statements like p and q or p and q and r then we would see the same results for all the rows of x and all the results of y that would be exactly the same results for all the rows that we see in the truth table. And what that means in sort of a non-technical sense is although x and y may be statements that look different from each other, that read in English different from each other, logically they always mean the same thing. So logical equivalence is a really powerful idea for us because in working with statements, improving statements, we're going to be able to rewrite one statement in another form that might be easier to understand but has the same logical value. So logical equivalent statements, we write them as x is logically equivalent to y. That's not quite an equal sign. It's actually stolen from geometry for like congruence, like triangle congruence. So we don't mean two statements are the same but we mean that they have the same truth values, the same logical values in all cases. Three important examples that you saw in these, that should make common sense to you on some level, not not p is logically equivalent to p. If I say it is not not raining outside then it must be that it's raining outside. Two other examples very important here. We're going to revisit these in later screencasts. p implies q is logically equivalent to not q implies not p. So this gives us another way to rewrite a conditional statement in a slightly different form. And very importantly here the negation of a conditional statement. If I want to negate the statement p implies q, it's actually not a conditional statement at all. It's p and not q. So in your preview activities again, the way that you determine these logical equivalences here was to make truth tables. You make a truth table for the left side, a truth table for the right side, and just check and make sure that they have the same truth values, either true or false, on for both of these statements in all the possible rows. So let's take a look at a simple logical equivalence here that involves by conditional statements. We're going to, we're going to show that not p if and only if q, that's one statement, is logical equivalent to p if and only if not q, another conditional statement. So what we're going to do is make a truth table here for the left hand side and for the right hand side of that proposed equivalence and just see that I get the same truth values in all possible rows here. So let's do this. This first column I've got not p, so that's going to be false, false, true, true. And here I need to write down not p if and only if q. Remember by conditional statements are true whenever the two statements joined by the double headed error have the same truth value. So I'm focusing on these two columns here that will be false, true, true, false. Okay, so keep your eye on that. That's the left hand side. Over here I need to get not q. That's going to be false, true, false, true. And then I need to look at p if and only if not q. So I've got to blank out a large portion of this truth table and I would have false, true, true, false. Unblank that and that's the right hand side. And the thing to notice is all the truth tables in those rows are the same as all the truth values in those rows. So that does actually prove those two statements are logically equivalent. Now let's move on to a concept check before we look at a more complicated example with three statements. We're just going to test your intuition on this. So take a look at the statement p implies q or p implies r. Now look at the following five statements and I want you to think about what should this statement be logically equivalent to? What do you think this is logically equivalent to? Is it p implies q, p implies r, p implies q or r, p implies q and r, or finally p or q implies r. Think it through, make some English examples up while you pause the video and when you come back we'll check the answer. Okay, so the answer here is going to be c, p implies q or r. Let's take a look and prove this. So we're going to make a truth table for the left side here, this entire or statement and then make a truth table for the right hand side. There's so much stuff that I'm going to do the left side on one slide and the right on another. So we have eight rows to consider here because we have three statements. So let's look at them one by one. Looking at the left hand side now that's what we're eventually going to come up with. P implies q, this is just looking at p and q so that's true, false, true, true, true, false, true, true. This would be p implies r so I got to look at p and r so that would be true, true, true, true and then false, false, true, true because in those last two cases the hypothesis p is false and now I need to take these two columns and join them with an or so as long as one or the other or both of these statements is true I have a true statement here. So it's true almost in every row except the sixth one right there. There's the loan of false in this truth table. Okay, so all true except for that sixth row. That's what the left hand side looks like. Now let's take a look at the right hand side and build that. So I'm going to build the q or r first. So that would be true, true, true, true, true, I'm sorry, yeah true, false, true and false. It's easy to get turned around in these things. So that's those two rows we had false for where both the statements were false. Now I need to look at p implies q or r. So there's the hypothesis. Here's my conclusion. So true, true, true, conditional statements are always true except in one case and here it's coming up here. That's when the hypothesis is true but the conclusion is false. There's the one place where I have, I have the false there. Now if we just hold on to this and back up a slide, that's exactly the same result as the left hand side. So the left hand side shakes out like this. The right hand side shakes out exactly the same way. Therefore those two statements are actually logically equivalent as we said they should be. And we just page back to the concept check. It's fairly easy to see that these two can't be logically equivalent to this but some of these others you might have guessed incorrectly here like this one and this one actually don't work but you might wonder why. So let's take a look very quickly at the results of the truth table for e just for an example. So I've made up the entire truth table here to show that p implies q or p implies r is not logically equivalent to p or q implies r. Let's just take a look at how the truth tables work. I'm not going to construct the truth tables. You should do that on your own. But here are the results. And again here's the results that we got in the previous work. And if you make the truth table for p or q implies r, what you notice here are two differences. Okay, in that fifth row and in the seventh row there's situations where p and q and r are a certain configuration. But the first statement ends up being true but the second one is false. So that means that they are not always logically the same. A lot of times they are but not always because they don't have the same truth values in all rows for all configurations of the variable. Those two statements are not logically equivalent. Thanks for watching.