If the opposite is impossible, the statement is true: To prove a statement by contradiction: assume it is false. Reason logically until you reach something impossible — a contradiction.
Since assuming it false led to nonsense, the assumption must be wrong, so the statement is true.
IB-style question — √2 is irrational
Prove that √2 is irrational.
Step by step
- Assume the opposite: √2 is rational, so √2 = p/q in lowest terms (p, q integers with no common factor, q ≠ 0).
- Square and rearrange.
- So p² is even ⇒ p is even. Write p = 2m.
- So q² is even ⇒ q is even. But then p and q are BOTH even — they share a factor 2.
- That contradicts 'lowest terms'. So the assumption is false: √2 is irrational.
Final answer
√2 cannot be written as p/q in lowest terms, so it is irrational.
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Same idea: suppose not, find the clash: The pattern is always: suppose the conclusion fails, follow the logic, and reach a statement that can't be true. The clash proves your supposition was wrong.
IB-style question — n² even ⇒ n even
Prove that if n² is even, then n is even (n ∈ ℤ).
Step by step
- Assume the opposite: n² is even but n is ODD.
- Square it.
- That's an odd number — but we were told n² is EVEN. Contradiction.
- So n cannot be odd: n is even.
Final answer
Assuming n odd made n² odd, contradicting n² even — so n is even.