Converses and contrapositives

Some of the statements we have met above have what is known as a converse. We shall start this section by giving you the converses of a number of statements. Have a look at each example and try to work out how you think we form the converse of a statement. Here are the examples:

Statement	Converse
if a and b are odd, then ab is odd	if ab is odd, then a and b are odd
if a and b are even, then ab is even	if ab is even, then a and b are even
if a is even, then $a^2$ is even	if $a^2$ is even, then a is even
if a is odd, then $a^2$ is odd	if $a^2$ is odd, then a is odd
if a and b are even, then $a + b$ is even	if $a + b$ is even, then a and b are even
if a and b are odd, then $a + b$ is odd	if $a + b$ is odd, then a and b are odd

Now we shall set out a table of the converses that are relevant to this specification:

Statement	Converse
if A then B	if B then A
A only if B	B only if A
A if B	B if A
A iff B	B iff A

From this table you can see that the converse of a statement is constructed by “swapping” A with B. We have already examined the truth tables of each of the above statements and their converses in earlier sections and we concluded:

• if A then B and its converse if B then A do NOT say the same thing: they are NOT equivalent statements

• A only if B and its converse B only if A do NOT say the same thing: they are NOT equivalent statements

• A if B and its converse, B if A do NOT say the same thing: they are NOT equivalent statements

• A iff B and its converse B iff A do say the same thing: they are equivalent statements

We can rewrite the table to include logically equivalent statements:

Statement	Converse
if A then B	if B then A
A only if B	B only if A
B if A	A if B
A iff B	B iff A
B iff A	A iff B

We have learnt that a statement and its converse are not the same logically. A natural follow-up question to ask is what statements are there that are logically the same as those we have met – the answer is found in the contrapositive of a statement. We shall start by listing the contrapositives that are relevant to this specification, and then we shall examine them in a little more detail:

Statement	Contrapositive
if A then B	if not B then not A
A only if B	not B only if not A
A iff B	not B iff not A

In each of these, A and B are both swapped and negated, whereas in the converse, they were simply swapped. (There is also a third possibility, called the inverse of a statement, where A and B are both negated, but not swapped. We will not consider inverses further here, and we do not test them in the TMUA.)

We examine the truth tables for each of these in turn:

A	B	if A then B	not B	not A	if not B then not A
T	T	T	F	F	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

From this we can see that if A then B and its contrapositive, if not B then not A, are logically equivalent statements.

And also:

A	B	A only if B	not B	not A	not B only if not A
T	T	T	F	F	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

From this we can see that A only if B and its contrapositive, not B only if not A, are logically equivalent statements.

Here are a few practical examples of statements and their contrapositives – for each one check you can see why they are equivalent statements and look carefully at how not is used in some of the examples:

if $x = 2$ then $x^2 = 4$ , if $x^2 \neq 4$ then $x \neq 2$

if $x^2 < 4$ then $x < 2$ , if $x \geq 2$ then $x^2 \geq 4$

if two triangles have the same angles as each other then they are similar

if two triangles are not similar, then they do not have the same angles as each other

Before you read on, make sure you have completed Exercise K questions 4 and 5 above.

It is important to take care when working out the contrapositive of complicated statements. Consider the statement (about integers) if $a$ and $b$ are odd, then $ab$ is odd. This statement is true. Its contrapositive is therefore also true. But what is it? It is very tempting to insert a careless ‘not’ to produce if $ab$ is not odd then $a$ and $b$ are not odd, and from this it is a short step to if $ab$ is even then $a$ and $b$ are even. However, this is false [why?], and it is not the contrapositive. The correct form (in English that makes sense and is not strangulated) is:

if $ab$ is even then $a$ and $b$ are not both odd

or

if $ab$ is even then at least one of $a$ and $b$ is even

The reason for this mistake is, of course, that negation is not as simple as it seems – here we need to use not (A and B) is logically equivalent to not A or not B.

Summary: converses and contrapositives

Often TMUA questions will give you a statement and ask about the truth values of the various related statement, such as the converse and contrapositive. You should remember that a statement and its contrapositive are logically equivalent, whereas this is not necessarily the case for a statement and its converse. To determine the truth or falsity of the converse of statement usually takes some extra effort.

We recommend you have a look through past TMUA papers and examine every question where the terms converse or contrapositive arise. You can look at the model answers too if you are not sure how to answer any question you identify.