CONTRAPOSITIVE AND CONVERSE

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.”

DEFINITION converse
The converse of the sentence   ‘If [beautiful math coming... please be patient] $\,A\,$, then $B\,$’   is the sentence   ‘If $\,B\,$, then $A\,$’.

Note that the converse switches the hypothesis and conclusion.

DEFINITION contrapositive
The contrapositive of the sentence   ‘If [beautiful math coming... please be patient] $\,A\,$, 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:

[beautiful math coming... please be patient] $A$ $B$ 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 $A\,$
TTFFTTT
TFFTFFT
FTTFTTF
FFTTTTT

An analysis of these truth tables shows the following:

‘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 [beautiful math coming... please be patient] $\,x\,$, 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.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Proof Techniques


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 13; there are 13 different problem types.)