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Blast into Math! Pure mathematics: the proof of the pudding is in the eating

• Hint for #1: This statement is of the form if A then B, which we can write succinctly as

A =⇒ B.

Remember, the converse reverses the direction of the arrow.

• Example for #2: This statement is also of the form if A then B. To form its contrapositive we
need to first negate both A and B. Let’s do this with the following statement: if it is good,
then it sparkles. In our example A is “it is good.” The negation of this is “it is not good.”
The statement B is “it sparkles,” and its negation is “it does not sparkle.” So in our example,
“not A” is “it is not good,” and “not B” is “it does not sparkle.” The second step to form the
contrapositive is to reverse the direction of implication, so that it goes from “not B” towards

“not A.” Therefore the contrapositive of our example is: “it does not sparkle implies it is not
good,” which is fun to say American style: if it don’t sparkle, then it ain’t good!
• Example for # 4: let’s negate the statement “every number is even.” The negation of “every”
is “not every.” So, the negation is “not every number is even.” This is equivalent to the
statement, “there exists a number which is not even,” because “not every number is even”
means that there must be some number which is not even.
• Example for # 5: let’s negate the statement “Bonn is always sunny.” If this is not true, it

means that Bonn is not always sunny, and that’s precisely the negation: Bonn is not always
sunny.
• Example for #6: let’s make the converse of the statement “if the sum of two numbers is
not even, then one of them is not even.” To form the converse we reverse the direction
of implication. Remember, the implication always goes from “if” towards “then.” In this
example the statement is if A then B, where A is “the sum of two numbers is not even” and
B is “one of them is not even.” So, the converse is: if one of two numbers is not even, then
their sum is not even.
• Example for # 7: let’s make the contrapositive of the statement “when it doesn’t rain, then

it doesn’t pour.” To do this we should put the statement in the form A implies B. In this
example the statement means “when it is not raining, then it is not pouring,” which means:
“if it is not raining, then it is not pouring.” In our example A is “it is not raining,” and B is
“it is not pouring.” To form the contrapositive we negate these. So, “not A” is “it is raining,”
and “not B” is “it is pouring.” Finally, the direction of implication in A =⇒ B is reversed:
the contrapositive is not A =⇒ not B. So, in this example the contrapositive is: it is pouring
implies it is raining, or in somewhat better English, if it is pouring, then it is raining!

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