Introduction

In the previous chapter on logical operators, we saw the implication operator, denoted by the symbol ::\to::.

Implications are fundamental to the study of mathematical proofs. Nearly all mathematical theorems make assertions in terms of implications. The premise(s) specifies the conditions of the theorem while the conclusion gives the final result.

For this reason, implications have been studied extensively throughout logic. In this chapter, we consider three other implications that could be created from a given implication and their practical importance in logic, as well as mathematics.

Converse

Let's say we have been given an implication ::p \to q::. One way in which we could change this implication is to switch the positions of the premise and the conclusion. This gives us the following.

::\pmb{q \to p}::

This new implication is called the converse of the original implication.

Let's consider an implication expressed purely in English, and then have a look over its converse:

If it is raining, then the cricket match is not being played.

What would the converse be for this implication? Well, just switch the premise and the conclusion and you're done!

If the cricket match is not being played, then it is raining.

This here is the converse of the original statement. Or if this statement was the original implication, then the previous one would be its converse.

Now let's try to reason completely naturally and intuitively whether the converse and the original implication say the same thing.

Here's what the original implication means.

If it's raining, then it's confirmed that the match is not being played. But there's nothing asserted if it is not raining. It might be or not be the case that the match is being played. For instance, it might not be raining, but the stadium's lights might be down because of which the match would be delayed.

The statement only states that if it is raining, then we know for sure that the match is not being played.

Let's now consider the converse.

It is saying that if the match is not being played, then it's confirmed that it is raining. The statement rules out the possibility of any other reason for the delay of the match. If no match is happening, then we know for sure that it can only be due to a rain.

But the original statement implicitly says that there might be some other reason as well for the match's delay, not just the fact that it's raining. However, if it's raining then certainly the match is not being played.

In the converse, if the premise if false i.e the match is being played, then it might be the case that it is raining. This clearly contradicts the original implication and therefore, intuitively, we reason that the converse of an implication is not equivalent to it.

But let's see whether mathematical logic can testify this natural reasoning of ours.

::p::::q::::p \to q::::q \to p::
::\bold T::::\bold T::::\bold T::::\bold T::
::\bold T::::\bold F::::\bold F::::\bold T::
::\bold F::::\bold T::::\bold T::::\bold F::
::\bold F::::\bold F::::\bold T::::\bold T::

As is apparent, both the implications are not equal to one another when the premise of one is true, while its consequent is false.

Inverse

The inverse of an implication is obtained by negating both the premise and the consequent. That is, if the original implication is ::p \to q::, then its inverse is given as:

::\pmb{\neg p \to \neg q}::

As before, time to reason about the inverse...

We'll consider the same original implication as before.

If it is raining, then the cricket match is not being played.

To understand what it says, you could refer to the section above. Anyways, let's now see the inverse.

If it is not raining, then the cricket match is not being played.

Here we are saying that if it is not raining, then for sure the match is being played. There just could be no other reason for the delay of the match. If there is no rain, we expect the match to be happening.

But what if it is raining? Then both possibilities are applicable: the match is being played or it is not being played. In either case, we couldn't refute this inverse implication. But, as we know, in the former case (when it is raining and the match is being played) the original implication becomes void.

So it turns out that both these implications don't seem to be equivalent. Do they?

Let's check it all mathematically:

::p::::q::::p \to q::::\neg p::::\neg q::::\neg p \to \neg q::
::\bold T::::\bold T::::\bold T::::\bold F::::\bold F::::\bold T::
::\bold T::::\bold F::::\bold F::::\bold F::::\bold T::::\bold T::
::\bold F::::\bold T::::\bold T::::\bold T::::\bold F::::\bold F::
::\bold F::::\bold F::::\bold T::::\bold T::::\bold T::::\bold T::

Once again, we can see that the inverse of the original implication is not equivalent to it.

Time for one more implication...

Contrapositive

When we do both the things of the converse and the inverse, we get the contrapositive for the original implication.

That is, the contrapositive is obtained when we switch the positions of the premise and the conclusion and then negate them. Hence, given the implication ::p \to q::, its contrapositive is:

::\pmb{\neg q \to \neg p}::

As before, let's demystify the relation of a contrapositive with the original implication by using an intuitive English sentence.

The original implication is just as before, and is detailed above:

If it is raining, then the cricket match is not being played.

The contrapositive of this statement is defined as follows:

If the cricket match is being played, then it is not raining.

What it states is that if the match is being played, then it is not raining for sure. If the premise is false i.e when the match is not being played, then it may or may not be raining.

See how similar this sounds to the original implication which asserted that there are other possible reasons for the match not happening such as the stadium's lights going down, and as you can verify this statement also asserts that fact.

Altogether, it seems that the contrapositive goes well with the original implication. Let's check it out using a truth table.

::p::::q::::p \to q::::\neg p::::\neg q::::\neg q \to \neg p::
::\bold T::::\bold T::::\bold T::::\bold F::::\bold F::::\bold T::
::\bold T::::\bold F::::\bold F::::\bold F::::\bold T::::\bold F::
::\bold F::::\bold T::::\bold T::::\bold T::::\bold F::::\bold T::
::\bold F::::\bold F::::\bold T::::\bold T::::\bold T::::\bold T::

Expectedly, both these implications are equal to one another in every valuation and hence express the same proposition.

Practical application of these

At this point, you might be thinking as to what is the practical-level use of these variations of a given implication. Well it turns out that the most important result is of the contrapositive.

When we are proving a mathematical assertion which assumes a premise, often times it might be difficult to prove the statement by going in the specified manner of first assuming the premise and then doing some math to arrive at the conclusion.

What might be way more intuitive would be to consider the contrapositive. That is, we assume the negation of the conclusion and arrive at the negation of the premise. If this could be proven, then our original implication gets proven automatically, since the contrapositive is equivalent to the original implication.

Below we show a very elementary example. More formal ways of dealing with such statements would be explored later on in this course.

Suppose you were asked to prove the following statement:

If the square of an integer is odd, then that integer is odd as well.

How would you approach such a problem logically and mathematically?

First of all, we know that integers are either odd or even (not odd). Secondly, there is a general form to express an odd integer — it is a multiple of 2, plus 1.

More succinctly, we could express an odd integer ::o:: as follows:

::o = 2k + 1 \space \text{(for some integer k)}::

Coming back to the statement to be proven, there is one obvious way to solve it. That is to assume the premise is true, and then show that the conclusion holds as well. Let's try this out.

We'll take the integer as ::x::. Likewise the premise says that:

::x^2 = 2k + 1::

Now we just have to show that ::x = 2s + 1::, for some integer ::s::. How to solve this heck of a mathematical problem.

As you'd expect the actual statement doesn't sound intriguing but precisely proving it directly is indeed intriguing. What else can we do?

Well, since an implication is equivalent to its contrapositive, why not prove that statement instead, if we could. If we succeed in doing so, then we would automatically have proven the original statement.

Let's go on and first convert the original implication into its contrapositive. The premise and conclusion change positions and get negated.

If an integer is not odd, then its square is not odd as well.

Since 'not odd' simply means 'even', we could simplify this statement to the following:

If an integer is even, then its square is even as well.

Now can we prove this?

Assuming the premise, we get the following:

::x = 2k::

Now we only need to show that ::x^2 = 2s::, for some integer ::s::, in order to prove the contrapositive.

Let's algebraically work on the premise and see where we end up. Squaring both sides, we get:

??\begin{aligned} x &= 2k \\ x^2 &= (2k)^2 \\ x^2 &= 4k^2 \\ x^2 &= 2(2k^2) \\ x^2 &= 2s \end{aligned}??

for ::s = 2k^2::. Since ::k:: is an integer, it turns out that ::s = 2k^2:: is also an integer, and hence we've shown that :::: for some integer ::s::. This proves the contrapositive statement, which in turn proves the original implication.

Contrapositive to the rescue!

Moving on

Now if you had a hard time understanding this proof, don't worry. This chapter wasn't meant to introduce you to even the simplest of proofs. Rather, it was meant to introduce to you the idea of converse, inverse and contrapositive of an implication and just show you an insight as to what is the practical importance of the contrapositive.

Later on in this course, we'll learn the skills to prove mathematical assertions by considering many of them and showing how to approach the proofs rigorously using contraposition, wherever applicable.