Converse, Inverse and Contrapositive

Every conditional has three relatives, made by swapping and negating its parts. Only one of them says the same thing as the original. This builds on Logical Equivalence.

The conditional family.
Name	Form	Equivalent to $p \to q$?
Original	$p \to q$	yes
Converse	$q \to p$	no
Inverse	$\neg p \to \neg q$	no
Contrapositive	$\neg q \to \neg p$	yes

Key idea

A conditional is equivalent to its contrapositive $\neg q \to \neg p$, but not to its converse $q \to p$ or its inverse $\neg p \to \neg q$. (The converse and the inverse are equivalent to each other.)

Why it works

$p \to q$ is false in exactly one row: $p$ true, $q$ false. The contrapositive $\neg q \to \neg p$ is false when $\neg q$ is true and $\neg p$ is false, which is the same row. So they have the same column. The converse $q \to p$ is false in a different row, $q$ true and $p$ false, so it makes a different claim. "If it is a dog, it is a mammal" is true; its converse "if it is a mammal, it is a dog" is not.

The original and the contrapositive match; the converse differs in two rows.

Worked examples

Worked example: All three relatives

Write the converse, inverse and contrapositive of $r \to \neg s$.

Converse (swap): $\neg s \to r$.
Inverse (negate both): $\neg r \to \neg \neg s$, which may be written $\neg r \to s$.
Contrapositive (swap and negate): $\neg \neg s \to \neg r$, that is $s \to \neg r$.

Answer: The contrapositive $s \to \neg r$ is the only one equivalent to the original.

Worked example: In English

"If you have a ticket, you get into the show." What is the contrapositive?

Swap the parts and negate both.

Answer: "If you do not get into the show, you do not have a ticket."

Common mistake

Treating the converse as equivalent. That is affirming the consequent in disguise: from "if it rained, the street is wet" you cannot get "if the street is wet, it rained", since a sprinkler could have wet it. The truth table shows the row ($q$ T, $p$ F) where the two disagree.

Worked example: A converse that fails

"If a shape is a square, it has four sides." Is the converse true?

The converse is "If a shape has four sides, it is a square."
A rectangle that is not a square has four sides.

Answer: No. The original is true and its converse is false, so they cannot be equivalent.

There is a neat pattern in the table: the inverse is the contrapositive of the converse. Swap $q \to p$ and negate both parts and you get $\neg p \to \neg q$. That is why the converse and the inverse always agree with each other, while the original and the contrapositive agree with each other.

The contrapositive is useful in proofs. When $p \to q$ is hard to show directly, showing $\neg q \to \neg p$ proves the same thing. Later this becomes a rule of replacement, Transposition (Transposition and Material Implication (Trans, Impl)).

Converse, Inverse and Contrapositive

Worked examples

Practice

Builds on

Leads to

Sources and further reading