 Welcome back to another screencast. This one is on converses and contrapositives. So the converse and contrapositive are statements that we form based on a given statement, one that is a conditional statement. So given the conditional statement P implies Q or if P then Q, the converse of that statement is a new conditional statement Q implies P. So the converse is a new conditional statement that we get by taking the original and just simply switching the hypothesis and conclusion. The contrapositive of the statement P implies Q is similar, except we're not only going to switch the hypothesis and conclusion, we're going to negate each of those. So it's not Q implies not P. So let's take a look at some examples here where we're forming the converse and contrapositive. So here are two sentences and we're going to form their converses first. So here are two conditional statements and we're going to form the converse of each. First one says if it's raining outside then I'll carry my umbrella. So the converse here, again we're just simply going to switch the if and the then here. So if I am carrying my umbrella, and I'll just write umbrella instead of writing all that stuff out, if I am carrying my umbrella then it's raining outside. There's the converse, very simple, but one thing you can tell fairly quickly is that the converse of a statement is not logically equivalent to the original statement. If it's raining outside I'll carry my umbrella but it doesn't mean that every time I carry my umbrella then it must be raining. Those two statements just do not mean the same thing. They're not logically equivalent. So the second statement says if P is a prime number and greater than two then P is odd. So let's form the converse of that. Very simply we're going to switch the hypothesis and conclusion. So we're going to say if P is odd then P is prime P is prime and P is greater than two. Right, so here again we see that this is not true even. This is not a true statement. If P is odd, it's not always the case that P is prime like P equals 15 for example. So those are the converses. Let's look and look at the contrapositives next. Same two statements before they're contrapositive. So here we're going to swap the hypothesis and conclusion but also negate each. So the contrapositive would say if I am not carrying my umbrella so that's how it would sound in English but to keep things short here I'm going to say if no umbrella or if not umbrella, if I'm not carrying my umbrella then it's not raining. Now this one actually seems like it could be logically equivalent to the original statement. If I'm not carrying my umbrella then how could it possibly be raining and that's actually that has actually the case because as we saw in an earlier screencast in an example, P implies Q is indeed logically equivalent to not Q implies P. And that was something you did in your preview activity. So very importantly, the most important thing you need to know about contrapositive is that they are always logically equivalent to the conditional statements that they came from. If you want to rewrite a conditional statement to get a better handle on what it's saying try rewriting the contrapositive instead. Let's look at this second statement here which gets a little complicated. So if P is prime bigger than two then P is odd. So we're going to swap the hypothesis and conclusion and negate each. So if P is not odd and I'm going to do something on the fly here, it would be okay to say if P is not odd but it would be nicer to say if P is even. That's what not odd means. Then, this one I have to think about, then it is not the case that P is prime and P is greater than two. Okay, so I've abbreviated, let me say it again in English because it's a little complex. If P is not odd, otherwise known as if P is even, then it is not the case that P is prime and P is greater than two. I would like to take this space up here just to clean that up a little bit. Starting with this piece right here, the new conclusion. I'm going to abbreviate the statement P is prime with the letter A and P bigger than two with the letter B. So what we're looking at here in the conclusion is not A and B. This conclusion in symbols, that's what it would look like. However, something important here in your reading and in part of the preview activities you've proven or will prove that not A and B is logically equivalent to not A or not B. That's a notion called de Morgan's laws. So I can rewrite the conclusion here and actually the entire contra positive by saying if P is even then this is going to change into an or statement. Then either P is not prime, that's what the not A reads as, or P is not bigger than two, which is to say that P is less than or equal to two. So both this and this up here are correct contra positives, but I think the one on top sort of reads a little bit more easily and the way we got there from the sort of straight raw contra positive is to use an additional rule about logical equivalences to replace stuff inside the statement we started with. Now let's end off with a concept check here. Here's a statement P implies Q or R. What is its contra positive? So you have to think about this for a little bit. Is it Q or R implies P? Is it not P implies not parenthesis Q or R? Is it not parenthesis Q or R implies P? Is it not parenthesis Q or R implies not P? Or finally is it not Q and not R implies not P? Lost a process so take your time pause the video and come back when you're ready. Okay so this was a little tricky of a question because there's actually two right answers to it that breaks the rules somewhat for these concept checks but you know two ways to be right here that would be both D and E. Now D is correct because that's the straight contra positive. I take the hypothesis and conclusion swap them and negate each. That's exactly what D is okay so that would be correct in a sort of straightforward way. E is also correct though because the two statements that are in the hypothesis here are themselves logically equivalent. Another application of De Morgan's law would say that the negation of Q or R is logically equivalent to not Q and not R. So if I replace one part of a statement with something that's logically equivalent to it, I have I haven't changed anything. Both entire statements are logically equivalent. So both of those are contra positives. You can rule out B and C fairly quickly. A here is also not the contra positive. A is the converse of the original statement. So the converse is right there. Okay so that's how to form converses and contra positives and use logical equivalence rules to help make them easier to read. Thanks for watching.