Demostracions

Demostracions are formal arguments used to establish mathematical truths. A demostració shows that a proposition P follows from axioms and previously proven results, through a sequence of justified steps. Each step relies on accepted rules of inference, definitions, and lemmas. A valid demostració does not rely on empirical observation; it aims to derive the conclusion logically from starting assumptions.

Structure and methods. A typical demostració begins with hypotheses and known premises, proceeds with a chain of deductions, and ends with the stated conclusion. Justifications accompany each step. Common methods include direct proof, proof by contradiction (assuming the negation of the conclusion and deriving a contradiction), proof by contrapositive, and proof by cases. Inductive proofs establish statements for all natural numbers, while constructive proofs provide explicit examples or constructions. Other approaches include the probabilistic method and non-constructive proofs, depending on the context and goals.

Examples and scope. Classic examples include showing that the sum of two even numbers is even, proving the irrationality of sqrt(2), or Euclid’s proof of the infinitude of primes. Demonstracions span the full range of mathematical disciplines, from elementary number theory to advanced analysis and topology. The level of detail and the style of justification may vary, but all valid demostracions meet the criteria of logical soundness within the chosen axiomatic framework.

Relation to logic and computation. Demonstracions are central to the reliability of mathematics: a proof provides certainty within a given system of axioms. In modern practice, proofs can be formalized in logical calculi and checked by computer-assisted proof systems such as Coq or Lean, which verify each inference step according to formal rules.

Notes. In Catalan, demostracions commonly refers to mathematical proofs, though the term can also denote demonstrations in other contexts; the mathematical sense emphasizes rigorous justification.