Converse and Equivalence

What you will get from this section. You will learn what the converse of an implication is, why the converse is not automatically true, and what it means for two statements to be equivalent.

Converse

Suppose we have an implication

P \Rightarrow Q.

The converse is the implication in the opposite direction:

Q \Rightarrow P.

In words, the converse of "if $P$ , then $Q$ " is "if $Q$ , then $P$ ".

It is very important to remember that an implication and its converse are not automatically both true. In general,

P \Rightarrow Q

does not automatically mean

Q \Rightarrow P.

Example 1

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

It is true that

n \text{ is a multiple of } 10 \Rightarrow n \text{ is even}.

But the converse is not true:

n \text{ is even} \Rightarrow n \text{ is a multiple of } 10.

For example, $n=6$ is even, but $6$ is not a multiple of $10$ .

Example 2

It is true that

x>2 \Rightarrow x^2>4.

But the converse is not true:

x^2>4 \Rightarrow x>2.

For example, $x=-3$ satisfies $x^2=9>4$ , but $x=-3$ is not greater than $2$ .

Example 3

It is true that

\text{a quadrilateral is a rectangle} \Rightarrow \text{its diagonals are equal in length}.

But the converse is not true:

\text{a quadrilateral has diagonals equal in length} \Rightarrow \text{it is a rectangle}.

For example, an isosceles trapezium has diagonals equal in length, but it is not a rectangle.

This example is slightly more subtle because equal diagonals are strongly associated with rectangles. However, equal diagonals alone do not force all angles to be $90^\circ$ , so they do not guarantee that the quadrilateral is a rectangle.

Equivalence

Sometimes an implication and its converse are both true. In that case, we say the two statements are equivalent.

So formally, $P$ and $Q$ are equivalent if both $P \Rightarrow Q$ and $Q \Rightarrow P$ are true. We write this as

P \Leftrightarrow Q.

This is read as " $P$ if and only if $Q$ ", or " $P$ is equivalent to $Q$ ", or " $P$ and $Q$ are equivalent".

Example 1

For an integer $n$ ,

n \text{ is a multiple of } 6 \Leftrightarrow n \text{ is a multiple of } 2 \text{ and } n \text{ is a multiple of } 3.

This is an equivalence because both directions are true.

If $n$ is a multiple of $6$ , then $n$ is a multiple of $2$ and $3$ .

Conversely, if $n$ is a multiple of both $2$ and $3$ , then $n$ is a multiple of $6$ .

Example 2

For an integer $n$ ,

n \text{ is odd} \Leftrightarrow n^2 \text{ is odd}.

This is an equivalence because both directions are true.

If $n$ is odd, then $n^2$ is odd.

Conversely, if $n^2$ is odd, then $n$ must be odd. This is because if $n$ were even, then $n^2$ would also be even.

Example 3

For a triangle with side lengths $a$ , $b$ and $c$ , where $c$ is the longest side,

\text{the triangle is right-angled} \Leftrightarrow a^2+b^2=c^2.

This is an equivalence because both directions are true.

If the triangle is right-angled, and $c$ is the hypotenuse, then by Pythagoras' theorem,

a^2+b^2=c^2.

Conversely, if a triangle has side lengths $a$ , $b$ and $c$ , where $c$ is the longest side, and

a^2+b^2=c^2,

then the triangle is right-angled.

So $a^2+b^2=c^2$ is not merely a consequence of being right-angled; it is also enough to guarantee that the triangle is right-angled, where $c$ is the longest side.

Example of a common mistake explained: squaring both sides does not always result in another equivalent equation!

When solving equations, students often square both sides. This can be useful, but it is not always reversible.

For example, consider

x=\sqrt{x+2}.

Squaring both sides gives

x^2=x+2.

So we have

x=\sqrt{x+2} \Rightarrow x^2=x+2.

But we do not automatically have

x^2=x+2 \Rightarrow x=\sqrt{x+2}.

Now solve the original equation carefully, using the correct implication symbol at each step:

x=\sqrt{x+2} \Rightarrow x^2=x+2.

Then

x^2=x+2 \Leftrightarrow x^2-x-2=0

\Leftrightarrow (x-2)(x+1)=0

\Leftrightarrow x=2 \text{ or } x=-1.

Notice that the first step used $\Rightarrow$ , not $\Leftrightarrow$ . This means the later statement $x=2 \text{ or } x=-1$ is only a consequence of the original equation, not necessarily equivalent to it.

So we must check the possible solutions in the original equation.

For $x=2$ :

2=\sqrt{2+2},

which is true.

For $x=-1$ :

-1=\sqrt{-1+2},

which is false, since $\sqrt{1}=1$ , not $-1$ .

Therefore, $x=-1$ is a spurious solution. It appeared because the first step, squaring both sides, in this case, was only a one-way implication, not an equivalence.

This does not mean that squaring both sides is always wrong. Sometimes squaring both sides does give an equivalent equation, and sometimes it only gives a one-way implication. This has to be judged case by case.

The wider lesson is that, when writing mathematics, we should be clear about exactly what each step means. Are we saying that the next statement is equivalent to the previous one, or only that it follows from the previous one? In other words, do we mean $\Leftrightarrow$ , or only $\Rightarrow$ ? These are not just symbols; they describe the logical relationship between the steps. Being careful with this distinction helps us avoid hidden mistakes in equations, proofs, and mathematical arguments more generally.

Summary

The converse of $P \Rightarrow Q$ is $Q \Rightarrow P$ .
An implication being true does not automatically mean its converse is true.
$P \Leftrightarrow Q$ means both $P \Rightarrow Q$ and $Q \Rightarrow P$ are true.
We read $P \Leftrightarrow Q$ as " $P$ if and only if $Q$ ".

Worksheets

Practise forming the converse $Q \Rightarrow P$ of an implication and testing whether it holds, and working with equivalence $P \Leftrightarrow Q$ — the "if and only if" relationship where an implication and its converse are both true. Each worksheet below comes as a PDF with fully worked solutions.

Worksheet 1 — Questions · Solutions