Leaving Certificate Mathematics Definitions Practice Test

What is the name of the proof that assumes a statement and then proves it wrong by using valid axioms?

Proof by contradiction Direct proof Inductive proof Empirical proof

Correct Answer:

Proof by contradiction

Explanation:

The main idea here is a proof by contradiction. You suppose that the statement you want to prove is false (its negation), and then you reason using valid axioms and logic to see what would follow. If this leads to a contradiction—something that cannot be true given the established rules—then the initial assumption must be false, so the original statement is true. This is also known as reductio ad absurdum in many contexts. Direct proofs go straight from the axioms to the conclusion without assuming the negation. Inductive proofs establish a claim for all natural numbers by proving a base case and showing that if it holds for one case, it holds for the next. Empirical proofs rely on observation and experimentation rather than formal deduction.

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