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MathidiaMathematics Academy
Proof and mathematical language
Lesson 4 of 75
Challenge

Contradiction and contrapositive

Objective. Choose and execute indirect proof strategies.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Proof by contradiction assumes the negation of the target statement and derives a logical impossibility, concluding the original statement must be true; proof by contrapositive instead proves ¬Q⇒¬P, which is logically equivalent to P⇒Q but can be easier to manipulate. The classic irrationality of √2 uses contradiction: assume √2=a/b in lowest terms, square to get a²=2b², deduce a is even, then b is even too, contradicting "lowest terms". Contrapositive suits statements like "if n² is even then n is even", proved instead as "if n is odd then n² is odd". Choosing between the two depends on which direction gives a more tractable algebraic deduction.

Worked example

Use proof by contradiction to show that there is no largest integer, assuming a largest integer N exists.

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  1. 1Assume for contradiction that N is the largest integer.
  2. 2Consider the integer N+1, which is also an integer since integers are closed under adding 1.
  3. 3Since N+1>N, N+1 is larger than the supposed largest integer N.
  4. 4This contradicts N being largest, so no largest integer exists.

Answer: No largest integer exists, since N+1>N always gives a contradiction

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Common mistake. Students confuse contradiction with contrapositive, assuming ¬P and ¬Q together (which proves nothing) instead of assuming ¬Q alone for contrapositive, or ¬(P⇒Q) alone for contradiction. Keep the two assumption structures distinct.

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