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.”
Note that the converse switches the hypothesis and conclusion.
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:
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 nicesounding way.
Let's illustrate with an example.
Consider this 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:
So, the answer you want is:
But, of course, you want to phrase it in the normal English way
(with the ‘for all’ implicit),
giving:
Got all that?
By the way, ‘for all’ sentences are studied in more detail in a future section,
Parallelograms and Negating Sentences.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
