Best Converse, Inverse, and Contrapositive Examples
This video shows examples of contrapositioning a sentence using the following three relationships: Converse (Q→P), Inverse (¬P→¬Q), and Contrapositive (¬Q→¬P).
Converse, Inverse, and Contrapositive
We’re going to look at these three terms using this statement, “If something is a dog, then it is a mammal.” Now if you take a second to think about that, that’s true. “Something’s a dog, it’s a mammal.” True. Yes, all dogs are mammals. I’m going to start with that, the fact that this statement is a true statement.
Now let’s look at this notation, this is read, “If P, then Q.” P is the part of your sentence (or the part of your statement) that comes after the word “if,” so “something is a dog” (this part right here) would be our P. Then Q is the part that follows “then,” or your conclusion, (it’s the end of your sentence) so that would be “it is a mammal,” this would be our Q here. Again, P is your hypothesis and Q is your conclusion.
If you wanted to write the converse of this statement the converse simply switches your P and your Q, or your hypothesis and your conclusion. To write the converse of, “If something is a dog, then it is a mammal,” we need to switch it around to say, “If something is a mammal, then it is a dog.”
Now, let’s think about that statement. This says, “If something is a mammal, it’s a dog,” so basically what it’s saying is, “All mammals are dogs.” Is that a true statement? Well, in order to prove it false, if you believe it’s false, (in order to prove it false) all you need is one counterexample, or one example of when this statement is not true.
One example of when this statement wouldn’t be true is a cat, a cat is a mammal and it’s not a dog; or a person, a person is a mammal, but a person is not a dog. This statement would be false, and we could give the counterexample of a cat—that’s our reason for that statement being false.
Let’s look at inverse. This little sign here (this notation) means, “Not P, not Q.” To write the inverse of our statement, we’re going to do “not P” and “not Q”, so we keep the same order, but we’re going to throw that word “not” in there. For this one it would be, “If something is not a dog, then it is not a mammal.”
Now let’s think about the truth of that statement, “If something is not a dog,” so let’s think of something that’s not a dog. “Not a dog” could be a cat, again, or a person, again. Then the end of this statement says, “Then it is not a mammal.”
Well our cat is not a dog, but it is a mammal, so that would mean that this statement, again, is false like our last one was, and we could, again, give the counterexample of a cat, because a cat proves that this statement is false. Our last term we’re going to talk about is contrapositive.
Contrapositive is, “Not Q, then not P.” Again, we’re taking our original statement and we’re reversing it, or flipping it around, like we did with the converse, but now it’s, “not Q” and “not P.” We’re starting with our Q, “It is a mammal,” but we’re going to do “not Q,” so, “If something is not a mammal, then it is not a dog.”
To the truth of the contrapositive “something is not a mammal,” so something that’s not a mammal would be a house, or a car, this marker is not a mammal; and it says, “If it’s not a mammal, then it’s not a dog.” Well, that’s true. Something that’s not a mammal is not going to be a dog, because it all dogs are mammals, so this is also true. If your statement is true, then your contrapositive should also, logically, always be true. These are converses, inverses, and contrapositives.