Contrapositive and Converse
Before studying this section, you may want to review:
Keep the following two examples in mind as you study this lesson. Consider the true implication:
If it is raining, then the ground is wet.
- Suppose the ground is wet. Is it necessarily raining? No—someone might have washed a car, and dumped a bucket of water on the ground.
- Suppose the ground isn't wet. Is it raining? Absolutely not—because if it were raining, then the ground would be wet, and it isn't.
The converse of the sentence
If $\,A\,,$ then $B$
is the sentence:
If $\,B\,,$ then $A$
Note that the converse switches the hypothesis and conclusion.
The contrapositive of the sentence
If $\,A\,,$ then $B$
is the sentence:
If $\,(\text{not } B)\,,$ then $(\text{not }A)$
Note that the contrapositive negates the conclusion, and makes it the hypothesis. It also negates the hypothesis, and makes it the conclusion.
Here are the truth tables for an implication, its contrapositive, and its converse:
| $A$ | $B$ | not $A$ | not $B$ |
an implication: If $\,A\,,$ then $B$ |
the contrapositive of the implication: If $\,(\text{not } B)\,,$ then $(\text{not }A)$ |
the converse of the implication: If $\,B\,,$ then $A$ |
|---|---|---|---|---|---|---|
| T | T | F | F | T | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | F |
| F | F | T | T | T | T | T |
An analysis of these truth tables shows the following:
-
An implication is equivalent to its contrapositive. Thus, the sentences ‘If $\,A\,,$ then $B\,$’ and ‘If $\,(\text{not } B)\,,$ then $(\text{not }A)$’ are completely interchangeable: if one is true, so is the other; if one is false, so is the other.
Mathematicians routinely find the contrapositive of an implication, to see if it is easier to work with than the original implication.
- An implication is NOT equivalent to its converse. Thus, the sentences ‘If $\,A\,,$ then $B\,$’ and ‘If $\,B\,,$ then $A\,$’ are not interchangeable. The truth of each sentence must be investigated separately.
‘If... Then...’ Sentences in English
When you start mixing English and mathematics, things can get a bit muddled. For example, many English ‘if... then...’ sentences are really ‘for all’ sentences in disguise, so you need to be a bit creative in phrasing the converses and contrapositives in a nice-sounding way.
Let's illustrate with an example. Consider this sentence:
If a creature is human, then it has a brain.
Lurking in the background is a universal set of creatures, where a human is one of many different types of creatures. Then, the sentence is really a shorthand for:
For all creatures, if a creature is human, then the creature has a brain.
Or, make it look a bit more math-like:
For all creatures $\,x\,,$ if $\,x\,$ is human, then $\,x\,$ has a brain.
The ‘for all creatures’ is implicit (not showing, but assumed to be there) in the normal English version of the sentence.
So, suppose you're being asked for the contrapositive of the sentence:
If a creature is human, then it has a brain.
Then you're really being asked for the contrapositive of the “if... then...’ part of the sentence:
For all creatures $\,x\,,$ if $\,x\,$ is human, then $\,x\,$ has a brain.
So, the answer you want is:
For all creatures $\,x\,,$ if $\,x\,$ doesn't have a brain, then $\,x\,$ isn't human.
But, of course, you want to phrase it in the normal English way (with the ‘for all’ implicit), giving:
If a creature doesn't have a brain, then it isn't human.
Got all that? By the way, ‘for all’ sentences are studied in more detail in a future section, Parallelograms and Negating Sentences.