The Contrapositive

What you will get from this section. You will learn what the contrapositive of an implication is, why it is equivalent to the original implication, and how to express it in several different but equivalent ways.

The contrapositive from necessary conditions

Suppose we have an implication

One way to say this is:

$Q$ is necessary for $P$ means that:

In other words:

Symbolically, we can write this as:

This implication is called the contrapositive of $P \Rightarrow Q$, and it is yet another way to express the idea of $P \Rightarrow Q$, in addition to all the other equivalent ways you saw in section 2.

Example 1

Let $P$ be "$n$ is a multiple of $10$", and let $Q$ be "$n$ is even".

We know that

Here, being even is necessary for being a multiple of $10$. Without being even, $n$ could not be a multiple of $10$.

So the contrapositive is

For an integer $n$, this can also be written as

Example 2

We know that

Here, $x>1$ is necessary for $x>2$. Without $x>1$, it is impossible to have $x>2$.

The contrapositive is

This is true. If $x$ is not greater than $1$, then $x$ certainly cannot be greater than $2$.

Example 3

Suppose

Being in England is necessary for being in London. Without being in England, Jane could not be in London.

So the contrapositive is

This is often how we naturally use the contrapositive in real life. If someone says, "Jane is not even in England", then we can immediately conclude that Jane is not in London.

The contrapositive family

So the contrapositive of $P \Rightarrow Q$ is $\neg Q \Rightarrow \neg P$, but $\neg Q \Rightarrow \neg P$ is itself an implication! So we can express it using the same family of equivalent phrasings we used before from section 2.

The following all express the contrapositive of $P \Rightarrow Q$:

• $\neg Q$ implies $\neg P$, or symbolically $\neg Q \Rightarrow \neg P$.
• If $\neg Q$, then $\neg P$.
• $\neg Q$ is a sufficient condition for $\neg P$.
• $\neg P$ is an inevitable consequence of $\neg Q$.
• $\neg P$ is a necessary condition for $\neg Q$.
• $\neg Q$ only if $\neg P$.

Example 4

Consider the implication

Here,

and

The contrapositive is

So the contrapositive family is:

• $n$ is not even implies $n$ is not a multiple of $6$.
• If $n$ is not even, then $n$ is not a multiple of $6$.
• $n$ is not even is a sufficient condition for $n$ not being a multiple of $6$.
• $n$ not being a multiple of $6$ is an inevitable consequence of $n$ not being even.
• $n$ not being a multiple of $6$ is a necessary condition for $n$ not being even.
• $n$ is not even only if $n$ is not a multiple of $6$.

For an integer $n$, "not even" could also be replaced by "odd".

Example 5

Consider the implication

Here,

and

The contrapositive is

So the contrapositive family is:

• Jane is not in France implies Jane is not in Paris.
• If Jane is not in France, then Jane is not in Paris.
• Jane not being in France is a sufficient condition for Jane not being in Paris.
• Jane not being in Paris is an inevitable consequence of Jane not being in France.
• Jane not being in Paris is a necessary condition for Jane not being in France.
• Jane is not in France only if Jane is not in Paris.

The two families of equivalent ways to express $P \Rightarrow Q$

We now have two families of statements that all express the same logical idea.

The first family is the original implication family:

• $P$ implies $Q$, or symbolically $P \Rightarrow Q$.
• If $P$, then $Q$.
• $P$ is a sufficient condition for $Q$.
• $Q$ is an inevitable consequence of $P$.
• $Q$ is a necessary condition for $P$.
• $P$ only if $Q$.

The second family is the contrapositive family:

• $\neg Q$ implies $\neg P$, or symbolically $\neg Q \Rightarrow \neg P$.
• If $\neg Q$, then $\neg P$.
• $\neg Q$ is a sufficient condition for $\neg P$.
• $\neg P$ is an inevitable consequence of $\neg Q$.
• $\neg P$ is a necessary condition for $\neg Q$.
• $\neg Q$ only if $\neg P$.

Every statement in these two families is equivalent to every other statement in the two families. So we now have more than 10 different ways to express the idea of $P$ implying $Q$.

One key skill you need to develop is the ability to write out all these different equivalent forms, given any one version of $P \Rightarrow Q$.

Optional: proof that the contrapositive is equivalent to $P$ implies $Q$.

This part is optional, but it is useful to see this result formally proved. The proof also gives a first glimpse of proof by contradiction, which is our next topic, and uses a neat self-application idea. I hope you find it interesting!

First, suppose

We want to prove

Assume $\neg Q$ is true. We need to show that $\neg P$ is true. Just to be clear, our known facts here are: $P\Rightarrow Q$, and $\neg Q$ is true. Based on these two facts, we wish to show that $\neg P$ must be true.

Now, in any situation, either $P$ is true or $\neg P$ is true. Let's examine these in turn to see if each is possible.

Case 1: suppose $P$ is true, but then by the fact $P \Rightarrow Q$, we immediately deduce that $Q$ is true, and now $Q$ and $\neg Q$ are both true. This cannot be right, which is what we call a contradiction. Thus the conclusion here is that our supposition that "$P$ is true" is wrong.

Now if Case 1 is wrong, then the only remaining possibility, Case 2, that $\neg P$ is true, must be true, as exactly one of them is true in any given situation. Hence we now conclude that under these conditions, $\neg P$ must be true, and we have proven:

By the way, this is an example of proof by contradiction, which I will discuss in detail in the next section.

To finish proving $P \Rightarrow Q$ is equivalent to $\neg Q \Rightarrow \neg P$, we next must show the reverse direction, or the converse of the result we have just proved, is also true, that is: $\neg Q \Rightarrow \neg P$ implies $P \Rightarrow Q$. For this, we can simply use the result $(*)$ we have just proven! By setting $P$ in $(*)$ as $\neg Q$, and setting $Q$ in $(*)$ as $\neg P$, the result $(*)$ immediately becomes:

But $\neg(\neg P)$ is just $P$, and $\neg(\neg Q)$ is just $Q$. Therefore this becomes:

So we have now shown both directions:

and

Therefore,

and they are equivalent.

Summary

• The contrapositive of $P \Rightarrow Q$ is $\neg Q \Rightarrow \neg P$.
• The easiest way to understand the contrapositive is through necessary conditions.
• If $Q$ is necessary for $P$, then without $Q$, $P$ cannot be true.
• Therefore, $P \Rightarrow Q$ gives $\neg Q \Rightarrow \neg P$.
• The original implication and its contrapositive are equivalent.
• There are now two families of equivalent ways to express $P \Rightarrow Q$: the original implication family and the contrapositive family.