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Pure · Proof

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Proof by contradiction

Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)

Pure · Proof

Proof by contradiction

Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)

Why it works

In a proof by contradiction you assume the opposite of what you want to prove, then reason until you hit something impossible — a statement that contradicts a known fact or your own assumption. Since the reasoning was valid, the only thing that can be wrong is the assumption, so the assumption is false and the original statement is true.

The skeleton is always the same:
  1. Assume the negation of the claim.
  2. Reason by valid steps.
  3. Reach a contradiction (something that is false, or that conflicts with the
assumption).
  1. Conclude: the assumption was false, therefore the original claim holds.
The step students get wrong is step 1 — negating the statement. The negation of "mm is even" (for an integer) is "mm is odd". The negation of "for all xx, PP" is "there exists an xx for which PP is false". Get the negation right and the rest is ordinary algebra; get it wrong and the whole proof is aimed at the wrong target. And step 3 is essential: you must actually reach a contradiction — if you prove the result directly without one, that is a deduction, not a proof by contradiction.