The Contrapositive

Swap, then negate

The contrapositive of if P then Q does both moves. Swap the hypothesis and the conclusion, then negate each of them. The result is if not Q then not P.

From one conditional there are now four statements: the original, if P then Q; the converse, if Q then P; the inverse, if not P then not Q; and the contrapositive, if not Q then not P.

For "if n is a multiple of 4, then n is even", the contrapositive is "if n is not even, then n is not a multiple of 4". For a whole number, not even means odd, so it reads "if n is odd, then n is not a multiple of 4".

Why it says the same thing

A conditional is broken only when its hypothesis is true and its conclusion is false. The contrapositive is broken when "not Q" is true and "not P" is false, which means Q is false and P is true. That is exactly the one case that breaks the original.

Two statements that are broken in the same cases are true in the same cases, so a conditional and its contrapositive are logically equivalent. The table checks all four statements in every case.

All four statements in every case of P and Q. The original and the contrapositive are false only in case 2, P true and Q false. The converse and the inverse are false only in case 3, P false and Q true. Rows that match belong to equivalent statements.

The example

The original, "if n is a multiple of 4, then n is even", is true, because a multiple of 4 is 4k for a whole number k, and 4k = 2(2k) is even.

Its contrapositive, "if n is odd, then n is not a multiple of 4", is true for the same reason. Both statements say that no number is a multiple of 4 and odd at the same time, and 4k = 2(2k) shows that no such number exists.

The converse, "if n is even, then n is a multiple of 4", is false at n = 6. A statement and its converse can differ; a statement and its contrapositive cannot.

The gold circle holds the multiples of 4, inside the circle of even numbers. Move the number being tested anywhere among the whole numbers. The statement and its contrapositive are never crossed out, because no number is in the gold circle and outside the even one. The converse and the inverse are crossed out together, by every number in the ring between the circles, such as 6.

Two pairs

So the four statements fall into two pairs. The original and its contrapositive are always true together or false together. So are the converse and the inverse, because the inverse is the contrapositive of the converse: swap the two parts of if Q then P and negate both, and you get if not P then not Q.

The two pairs need not agree with each other. For the multiples of 4, the first pair is true and the second is false. They can also both be true. "If a whole number is odd, then it leaves remainder 1 when divided by 2" is true, and so is its converse, so all four statements hold.

Proving the contrapositive instead

Because a statement and its contrapositive are equivalent, proving one proves the other. That helps when the negated parts are easier to work with.

To prove "if $n^2$n² is even, then n is even", starting from $n^2$n² is even gives little to work with. Prove the contrapositive, "if n is odd, then $n^2$n² is odd", instead. Write n = 2k + 1, where k is a whole number. Then $n^2=4k^2+4k+1=2\left(2k^2+2k\right)+1$n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is one more than an even number, so $n^2$n² is odd. That proves the original statement, with no extra step.

The usual mistakes

Doing only one of the two moves. Swapping without negating gives the converse; negating without swapping gives the inverse. The contrapositive of "if $n^2$n² is odd, then n is odd" is "if n is even, then $n^2$n² is even": both parts swapped and both negated.

Negating a part wrongly. The contrapositive of "if it is raining, then the match is off" is "if the match is on, then it is not raining". The negation of "the match is off" is "the match is on".

Expecting the converse to follow from the original. They are in different pairs, so the truth of one says nothing about the other.

The museum’s price board

In the application below, the board says that a visitor under 12 enters free. Under the museum’s full policy, the board and its contrapositive are true and the converse and the inverse are false: one pair each.