In this video, we're going to talk about how to write the converse, the inverse, and the contrapositive of a conditional statement. So what exactly is a conditional statement? What do you think it is? Now, you've seen this before.
And the simplest way to describe it is as follows. If P... then Q.
That is a conditional statement. So let's say if you live in Los Angeles then you live in California. That's an example of a conditional statement.
The part that comes after the if which is denoted by P, that is the hypothesis. The stuff that comes after Q is known as the conclusion. Now you need to be familiar with the symbols that is associated with negation. So this symbol means not P.
So for example, Let's say it is sunny outside. The negation of that statement is it is not sunny outside. So if you see this symbol, it just means it's not happening.
It's a negation. Now what about the converse? What is the converse of the original conditional statement? The converse is basically the reverse of the conditional statement.
So you just got to switch Q and P. So if Q, then P. That is the converse.
If the conditional statement is true, and if the converse is true, what you have is a biconditional statement. But the converse is not always true. Sometimes it's true, sometimes it's not. The next term you need to be familiar with is the inverse. Now the inverse is simply the negation of the conditional statement.
if not P then not Q so that's the inverse the last one needs to be familiar with is the contrapositive the contrapositive is basically the reverse negation of the conditional statement if not Q then not P and I'll give you some examples of that and finally we talked about the biconditional statement In a biconditional statement, what you have is, it occurs when the conditional statement and the converse have the same true value. So if the conditional statement is true, and if the converse is true, then you have a biconditional statement. In the conditional statement, if P then Q can be symbolized by this expression. In a biconditional statement, it can go both ways.
So, if P then Q or if Q then P. Both will be true or both will be false. That's a biconditional statement.
So, let's try an example. If you live in Los Angeles. Then you live in California.
So this is the conditional statement. Now what is the hypothesis and what is the conclusion? The hypothesis is the stuff that comes after the if part.
So the hypothesis is living in Los Angeles. The conclusion is living in California. Now how can we write the converse and the inverse in addition to the contrapositive of the statement? So right now, keep in mind, the hypothesis is P. So P is living in Los Angeles, and Q is living in California.
So right now we have if P, then Q. So that's the conditional statement. To write the converse, we need to recall that the converse is the reverse of the conditional statement. If Q... then P.
So what we need to do is switch the second part of the sentence with the first part. So therefore the converse will be as follows. If you live in California, then you live in Los Angeles. So that is the converse. Now is the converse a true statement or is it a false statement?
Right now we're assuming that the conditional statement is true, but the converse is a false statement. Someone who lives in California doesn't have to live in Los Angeles, they could live in Sacramento, they could live in San Francisco. San Diego so the converse is false so if you have a true conditional statement and a false converse statement then this is not a bi conditional statement if the converse was true and if the conditional statement assuming that's true as well then we would have a bi conditional statement now let's write the inverse of the conditional statement. So recall that the inverse is simply the negation of the conditional statement. So if not P, then not Q.
So the sentence will be as follows. If you don't live in Los Angeles, I'm going to write LA for Los Angeles, then you don't live... California so I'm gonna write CA for California so that's the inverse if you don't live in Los Angeles you don't live in California so you don't reverse the sentence you simply negate the sentence from being yes to no Now what about the contrapositive? Oh, by the way, is the inverse true or false? If you don't live in L.A., then you don't live in California.
What would you say? Well, just because you don't live in L.A. doesn't mean...
you don't live in California because you can live in San Francisco which is not in LA but still within the state of California so in this case the inverse is false it turns out that if the converse is false the inverse is going to be false those two have the same true value if the converse is true then the inverse is going to be true as well Now let's write the contrapositive statement. So remember, the contrapositive is basically the reverse negation of the conditional statement. And it says, if not Q, then not P. So we've got to reverse it and negate it. So the sentence will be as follows.
If you don't live in California, then you don't live in Los Angeles. Now, is that a true statement? It turns out that it is, because if you don't live in the state of California, you can't live inside a city within California, which is LA.
Now, it turns out that if the conditional statement is true, then the contrapositive will be true. And if the conditional statement is false, the contrapositive will be false. So the conditional statement and the contrapositive will always have the same truth value. They will always be the same.
In the same way as the converse and the inverse are always the same. Anytime you have a reverse negation of something, it's like a double reverse. It's going to take you back to the original value. And that's what's happening with a contrapositive. Not only do you reverse the statements, but you negate it as well, which makes it true again.
Go ahead and work on this example problem. If I am hungry, then I will eat pizza. So given that conditional statement, write the converse, the inverse, and the contrapositive. So let's start with the converse.
The converse is simply the reverse of the conditional statement. So, we could say, if I eat pizza, then I am hungry. So that's the converse. We just gotta reverse the sentence.
Now let's move on to the inverse. The inverse is simply a negation of the sentence. So if I am not hungry, then I will not eat pizza. So that's the inverse. You just got to negate the original sentence.
And finally, let's write the contrapositive, which is a reverse negation of the original sentence. So not only do we have to reverse it, we need to negate it as well. So if... If I don't eat pizza, or if I will not eat pizza, however you want to phrase it, if I don't eat pizza, then I am not hungry.
So that's the contrapositive.