Discrete! #3: Discrete Mathematics and Its Applications 1.1C

Converse, Inverse, Contrapositive!

So, remember last time we talked about the conditional p→q, which is pronounced “if p, then q.” In this section, we’re going to talk about some related conditionals, called converse, inverse, and contrapositive.

For the sake of clarity, let’s set up a p and a q:

p: Zach’s pants fall.

q: People gasp.

Now, let’s set up the conditional:

p→q: If Zach’s pants fall, then people gasp.

Now, let’s learn these cool variations. First, we’re going to talk about the converse. To get the converse, you flip the conditional. So, now you have this:

Converse:

q→p: If people gasp, then Zach’s pants fall.

You can readily see that the converse doesn’t follow from the initial statement. We know that when my pants drop, people gasp in amazement. However, it’s not clear that if people gasp, my pants will fall. It COULD be the case, but it doesn’t follow from the initial statement.

To get the inverse, you negate everything. So…

The Inverse:

¬p→¬q: If Zach’s pants do not fall, then people will not gasp.

Once again, you see this doesn’t follow from our initial statement. My pants could be up, and people might still gasp. For example, if I took off my shirt people would gasp in amazement, even though my pants were still secured in place. Once again, the inverse COULD be true. For example, I could have machines attached to everyone’s jaw that prevent them from gasping unless my pants go down. BUT, the inverse is not implied by the initial statement.

Lastly, we have the contrapositive, which you get by negating AND flipping.

The Contrapositive:

¬q→¬p: If people do not gasp, Zach’s pants are not dropped.

The contrapositive DOES follow from the premise. It may not be immediately obvious that this is true, but think about it: We stated at the beginning “If Zach’s pants drop, people gasp.” To double-emphasize the facts, we could say “Every time Zach’s pants drop, people gasp,” or “It is always the case that when Zach’s pants drop, people gasp.” These statements all mean the same thing.

If the people haven’t gasped, then at least one thing is certain: Zach’s pants have not dropped. If they had been dropped, people would be gasping.

 

So, to review:

If you flip p for q, that’s the converse.

If you negate p and q, that’s the inverse.

If you flip AND negate, that’s the contrapositive.

Only the contrapositive follows from the original conditional.

 

Next post: Biconditional curious?

 

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One Response to Discrete! #3: Discrete Mathematics and Its Applications 1.1C

  1. Juby says:

    Good entry! I would recommend adding in a counter example for the converse as well. Also, you might point out that the inverse and converse are equivalent as well.

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