The Converse and the Inverse

Swapping the two parts

Start from a conditional, if P then Q. Swap the hypothesis and the conclusion and you get its converse, if Q then P. In symbols, the original is $P\Rightarrow Q$P ⇒ Q and the converse is $Q\Rightarrow P$Q ⇒ P.

The converse makes a different claim. The original promises that whenever P holds, Q holds; the converse promises that whenever Q holds, P holds. One of these can be true while the other is false.

A true statement with a false converse

Take "if n is a multiple of 4, then n is even". It is true: a multiple of 4 is 4k for some whole number k, and 4k = 2(2k) is even.

Its converse is "if n is even, then n is a multiple of 4". That is false, and one counterexample shows it: n = 6. Six is even, so the hypothesis of the converse is true, and 6 is not a multiple of 4, so its conclusion is false. That is the one case that breaks a conditional.

Other even numbers that are not multiples of 4, such as 2 and 10, break the converse as well, but one counterexample is enough.

A converse is not always false. "If a whole number is odd, then it leaves remainder 1 when divided by 2" is true, and so is its converse, "if a whole number leaves remainder 1 when divided by 2, then it is odd". The point is that the truth of one never settles the other.

The gold circle holds the multiples of 4 and sits inside the circle of even numbers. Move the number being tested. In the gold circle, at n = 8, it breaks nothing. In the ring between the circles, at n = 6, it is even and not a multiple of 4, and it breaks the converse and the inverse together. Outside both circles, at n = 7, it breaks nothing again.

Negating both parts

The inverse negates both parts and keeps them where they were: if not P then not Q. For the multiples of 4 it is "if n is not a multiple of 4, then n is not even".

To negate a part, write the statement that is true exactly when that part is false. For a whole number n, the negation of "n is even" is "n is odd", so the inverse can also be read "if n is not a multiple of 4, then n is odd".

The same counterexample

The inverse is false too, and n = 6 breaks it. Six is not a multiple of 4, so the hypothesis of the inverse is true. Six is even, so its conclusion, "n is not even", is false.

This is not a coincidence. The converse, if Q then P, is broken when Q is true and P is false. The inverse, if not P then not Q, is broken when "not P" is true and "not Q" is false, which again means P is false and Q is true. The two statements are broken in exactly the same case, so they are true together or false together. The table checks all four cases.

Every case of P and Q, with the converse, if Q then P, and the inverse, if not P then not Q, worked out in each. Both are false only in case 3, where P is false and Q is true, so their rows match.

Why the original can differ

The original, $P\Rightarrow Q$P ⇒ Q, is false only in case 2, where P is true and Q is false. The converse is false only in case 3. Different cases break them, which is why the original can be true while its converse is false, as with n = 6.

The usual mistakes

Mixing up the two moves. The converse swaps the parts and negates neither; the inverse negates both parts and swaps neither. Doing both at once gives a third statement, the contrapositive, if not Q then not P.

Writing the inverse as the denial of the whole statement, "it is not true that every multiple of 4 is even". The inverse keeps the if-then form and negates each part.

Assuming that a true statement has a true converse. "If a shape is a square, then it has four sides" is true. Its converse, "if a shape has four sides, then it is a square", is false: a rectangle that is not a square has four sides.

A museum’s price board

In the application below, a board says that a visitor under 12 enters free, and the full policy also lets in visitors aged 65 and over. A visitor aged 65 breaks the converse and the inverse together, as n = 6 does here.