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.
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 $A\,$ |
| 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
[beautiful math coming... please be patient]
$\,A\,$, then $B\,$’
and
‘If
[beautiful math coming... please be patient]
$\,(\text{not } B)\,$, 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
[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