1.8: Converse and Contrapositive

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The converse of an implication ${A} \Rightarrow {B}$ is the implication ${B} \Rightarrow {A}$. For example, the converse of “if Tiana pays the cashier a dollar, then the server gives Tiana an ice cream cone” is “if the server gives Tiana an ice cream, then Tiana pays the cashier a dollar.” It should be clear that these are not saying the same thing. (For example, perhaps Tiana has a coupon for a free cone.) This illustrates the fact that the converse of an assertion is usually not logically equivalent to the original assertion. In other words (as was mentioned in Section 1.4), the connective $\Rightarrow$ is not commutative:

Exercise $\PageIndex{1}$

Show that $A\Rightarrow B$ is not logically equivalent to its converse $B \Rightarrow A$.

The inverse of an implication ${A} \Rightarrow {B}$ is the implication $\lnot{A} \Rightarrow \lnot{B}$. For example, the inverse of “if Tiana pays the cashier a dollar, then the server gives Tiana an ice cream cone” is “if Tiana does not pay the cashier a dollar, then the server does not give Tiana an ice cream cone.” It should be clear that these are not saying the same thing (because one assertion is about what happens if Tiana pays a dollar, and the other is about the completely different situation in which Tiana does not pay a dollar). This illustrates the fact that the inverse of an assertion is usually not logically equivalent to the original assertion:

Exercise $\PageIndex{2}$

Show that $A\Rightarrow B$ is not logically equivalent to its inverse $\lnot A \Rightarrow \lnot B$.

The contrapositive of an implication is the converse of its inverse (or the inverse of its converse, which amounts to the same thing). That is,

\[\text{ the contrapositive of } A\Rightarrow B\text{ is the implication }\lnot B\Rightarrow\lnot A\]

For example, the contrapositive of “if Tiana pays the cashier a dollar, then the server gives Tiana an ice cream cone” is “if the server does not give Tiana an ice cream cone, then Tiana does not pay the cashier a dollar.” A bit of thought should convince you that these are saying the same thing. This illustrates the following important fact:

\[\text{Any implication is logically equivalent to its contrapositive.}\]

Exercise $\PageIndex{3}$

Show that $A\Rightarrow B$ is logically equivalent to its contrapositive $\lnot B \Rightarrow \lnot A$.

Remark $1.8.4$.

The inverse will not be important to us, although the converse and the contrapositive are fundamental. However, it may be worth mentioning that the inverse is the contrapositive of the converse, and therefore the inverse and the converse are logically equivalent to each other.

Warning.

Implications (that is, those of the form ${A} \Rightarrow {B}$) are the only assertions that have a converse or a contrapositive. For example, the converse of “I hate cheese” does not exist, because this assertion is not an if-then statement.

Exercise $\PageIndex{4}$

State (a) the converse and (b) the contrapositive of each implication. (You do not need to show your work.)

If the students come to class, then the teacher lectures.
If it rains, then I carry my umbrella.
If I have to go to school this morning, then today is a weekday.
If you give me $5, I can take you to the airport.
If the Mighty Ducks are the best hockey team, then pigs can fly.
Alberta is a province.
If you want to do well in your math class, then you need to do all of the homework.

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Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Remark \(1.8.4\).

Warning.

Exercise \(\PageIndex{4}\)