Mathematical Proof by Contradiction

Proof by contradiction is a mathematical method used to prove a statement by assuming that the statement is false.

The opposite of the statement to be proved is assumed temporarily at the beginning of a proof by contradiction.

Logical reasoning is then used to show that the assumption leads to an impossible or contradictory result.

When the assumption leads to a contradiction, the original statement must be true.

A contradiction occurs when the reasoning produces a result that cannot be true.

Proof by contradiction is also known as reductio ad absurdum.

The structure of a proof by contradiction is: assume the opposite, reason logically, obtain a contradiction, and conclude that the original statement is true.

The initial assumption must be the correct logical negation of the statement being proved.

A contradiction may arise because a result conflicts with a known fact, definition, assumption or established theorem.

Proof by contradiction is commonly used to prove that certain numbers are irrational.

A classic example is the proof that the square root of 2 is irrational.

Proof by contradiction can also be used in arguments involving prime numbers.

The proof that there are infinitely many prime numbers can be constructed using contradiction.

Proof by contradiction may be useful when a direct proof is difficult to construct.

In number proofs, parity can help create contradictions involving even and odd integers.

An even integer can be represented as 2n, where n is an integer.

An odd integer can be represented as 2n + 1, where n is an integer.

When proving irrationality by contradiction, a number is often assumed to be rational and written as a fraction in lowest terms.

If the reasoning shows that both the numerator and denominator must share a common factor, this contradicts the assumption that the fraction was in lowest terms.

The contradiction itself is not the final conclusion; it shows that the original assumption must have been false.

After identifying the contradiction, the proof should explicitly state that the original mathematical statement is therefore true.

A proof by contradiction must use valid logical and mathematical steps throughout.

Finding an unexpected result is not automatically a contradiction unless it conflicts with something that is known or assumed to be true.

Proof by contradiction is different from proof by deduction because contradiction begins by assuming the negation of the required result.

Proof by contradiction is different from proof by exhaustion because exhaustion checks every possible case within a finite set.

Proof by contradiction is different from a counterexample because a counterexample is used to disprove a universal statement.

A common mistake is assuming the original statement instead of assuming its opposite.

Another common mistake is failing to identify exactly why the final result is contradictory.

A well-written proof by contradiction should clearly identify the assumption, reasoning, contradiction and conclusion.