Mathematical Language

Contrapositive Form of an "If ... then ..." Statement

The statement "If $(A)$ then $(B)$ " means that if the hypothesis $A$ is true, then the conclusion $B$ is guaranteed to be true.

Suppose we know that in a certain town the statement

"If (a building is a church) then (the building is green)"

is a true statement,. What can we validly conclude about a red building? Based on the information we have, we can validly conclude that the red building is "not a church" since every church is green. We can also conclude that a blue building is not a church. In fact, we can conclude that every "not green" building is "not a church." That is, if the conclusion of a valid "If ... then ... " statement is false, then the hypothesis must also be false.

The contrapositive form of "If $(A)$ then $(B)$ " is

"If (negation of $B$ ) then (negation of $A$ )" or "If ( $B$ is false) then ( $A$ is false)."

The statement "If $(A)$ then $(B)$ " and its contrapositive "If (not $B$ then (not $A$ )" are equivalent.

What about a green building in this town? The green building may or may not be a church – perhaps every post office is also painted green. Or perhaps every building in town is green, in which case the statement "If (a building is a church) then (the building is green)" is certainly true.

Practice 4: Write the contrapositive form of each of the following statements.

(a) If a function is differentiable then it is continuous.

(b) All men are mortal.

(d) If (2 divides $x$ and 3 divides $x$ ) then (6 divides $x$ ).