Proof by Contraposition vs. Proof by Contradiction

Two ways to argue backwards#

Take a claim everyone agrees with: “If it is raining, the ground is wet.”

Suppose you had to prove it. There are two backwards routes. They look similar and they are not.

Route one. Show that whenever the ground is dry, it is not raining. Prove that and you are finished — it says the same thing as the original. That is contraposition.

Route two. Suppose the claim is false. That would mean it is raining and the ground is dry, at the same time. Show that is impossible, and the claim must be true. That is contradiction.

Look at what you get to work with in each. Route one hands you one fact (dry ground) and a fixed destination (no rain). Route two hands you two facts (raining, dry ground) and lets you head anywhere impossible. That difference is the whole topic.

Both routes have one shared name. A proof that starts from the opposite of what you want, instead of marching straight at it, is called an indirect proof. Contraposition and contradiction are the two indirect methods; a proof that goes straight from the hypothesis to the conclusion is a direct proof. So “use an indirect proof” is asking for one of the two routes on this page. Usually contradiction. But contraposition is just as valid unless the question names one.

The difference, in one line#

Write the claim as p → q — “if p then q”, where p is the first half and q is the second. ¬ means “not”.

Contraposition up close#

Despite the name, there is nothing indirect happening. You are proving a different sentence — ¬q → ¬p — by an ordinary forward argument. You are allowed to, because that sentence means the same thing as the one you were asked about. (Why they mean the same thing is worked out in Converse, inverse, and contrapositive.)

Announce it in the first line. Say you are proving the contrapositive and say what it is. A reader who does not know why you opened by assuming the opposite of the conclusion will think the proof is running backwards.

Contradiction up close#

Here you do not swap the statement. You suppose the claim itself fails, and show the world breaks.

To deny “if p then q” you need p and ¬q together, because that is the only situation where an if–then is false. That is why you end up with two facts, and it is the practical advantage of the method. Then you reason until something impossible turns up. A number that is both even and odd. A fraction in lowest terms whose top and bottom share a factor. 0 = 1. Anything at all.

Name the contradiction. “This contradicts the assumption that a/b was in lowest terms” is a finished proof. “Contradiction!” with nothing named is where the marks disappear.

The same claim, proved both ways#

Claim. For every whole number n, if n² is even then n is even.

Here p is “n² is even” and q is “n is even”.

By contraposition. We prove: if n is odd, then n² is odd. Assume n is odd, so n = 2m + 1 for some whole number m. Then n² = (2m + 1)² = 4m² + 4m + 1 = 2(2m² + 2m) + 1, which is odd. Therefore if n² is even, n is even. ∎

By contradiction. Suppose n² is even but n is odd. Since n is odd, n = 2m + 1, so n² = 2(2m² + 2m) + 1 is odd. But we assumed n² is even, and no whole number is both. Impossible. Therefore n is even. ∎

The algebra is identical. The difference is entirely structural. The first proof never once mentions that n² is even — it simply arrives at “n² is odd” and stops. The second carries “n² is even” the whole way and uses it at the very end to collide with what it derived.

Notice which is cleaner. When the only contradiction you reach is “¬p contradicts p”, the extra assumption bought you nothing and contraposition was the better tool. That is the usual case for claims shaped like this one.

When to reach for contradiction#

Contradiction earns its keep in two situations.

• When you need both facts at once. Some arguments genuinely require p and ¬q on the table together to collide.
• When the claim is not an if–then at all. “The square root of 2 is irrational” and “there are infinitely many primes” have no first half to flip. Contraposition is not even defined for them; contradiction is the natural tool.

Spotting a mislabelled proof#

Read the first line and the last line, and ignore the heading.

• Opens by assuming only ¬q and ends at ¬p → contraposition, whatever it says at the top.
• Opens by assuming both p and ¬q and ends at something impossible → contradiction.
• Opens with p and ¬q but ends at “…which contradicts p” → this is a contradiction proof that should have been a contraposition. It works, but the second assumption was never used.

Where these go wrong#

• Assuming p during a contraposition. You get ¬q and nothing else. Bringing p along quietly turns it into a contradiction proof, whatever the heading says.
• Reaching something other than ¬p in a contraposition. The target is fixed. Landing on a generic absurdity means you drifted into the other method.
• Assuming only ¬q in a contradiction. The denial of p → q is p and ¬q. Dropping p throws away half of what you were entitled to.
• Not naming the contradiction. Write down which two things collide.
• Negating the conclusion carelessly. If q is “n is even and n > 2”, then ¬q is “n is odd or n ≤ 2” — the joining word flips, and the proof now needs both cases. Getting this wrong makes everything after it unfixable.
• Trying to contrapose something that is not an if–then. “√2 is irrational” has no first half.
• Confusing either one with proving the converse. Proving q → p is neither method — it is proving a different theorem.

Frequently asked questions#

Something here not clear? A topic you wish we covered? Tell us. We read every message, and a request is the fastest way to get a guide written — several of these exist because somebody asked.