motsatsbevis

Motsatsbevis, meaning "proof by contradiction" in Swedish, is a fundamental method of mathematical proof. It involves assuming the opposite of what you want to prove is true, and then showing that this assumption leads to a logical contradiction. Once a contradiction is established, it means the initial assumption must be false, thereby proving the original statement to be true.

This technique is often employed in various branches of mathematics, including number theory, set theory, and logic. For instance, proving that the square root of 2 is irrational is a classic example of motsatsbevis. One assumes that the square root of 2 can be expressed as a fraction p/q, where p and q are integers with no common factors. Through algebraic manipulation, this assumption leads to the conclusion that both p and q must be even, which contradicts the initial condition that they have no common factors. Therefore, the assumption that the square root of 2 is rational must be false, proving its irrationality.

The effectiveness of motsatsbevis lies in its ability to indirectly establish the truth of a statement. By demonstrating that the negation of the statement leads to an absurdity, the original statement is vindicated. This method requires careful logical deduction to ensure that the derived contradiction is a genuine inconsistency and not merely a misunderstanding of the assumptions.