Proof

For other uses, see Proof (disambiguation).

A proof is a logical argument that demonstrates the truth of a given statement or proposition. Proofs are essential in mathematics, logic, and the sciences, where they are used to establish the validity of theorems, hypotheses, and scientific theories.

Types of Proof

Proofs can be categorized into several types, each with its own methodology and application:

Direct Proof

A direct proof involves a straightforward application of logical steps to arrive at a conclusion. It starts with known facts and uses logical operations to derive the statement to be proven.

Indirect Proof

An indirect proof, also known as a proof by contradiction, assumes the negation of the statement to be proven and shows that this assumption leads to a contradiction, thereby proving the original statement.

Constructive Proof

A constructive proof not only demonstrates that a statement is true but also provides a method for constructing an example of the statement.

Non-constructive Proof

A non-constructive proof shows that a statement is true without necessarily providing a concrete example.

Proof by Induction

Mathematical induction is a method of proof used primarily in number theory and combinatorics. It involves proving a base case and then showing that if the statement holds for an arbitrary case, it holds for the next case.

Proof by Exhaustion

Proof by exhaustion involves checking all possible cases to ensure that a statement holds in each one.

Importance of Proof

Proofs are fundamental to the advancement of knowledge in various fields. In mathematics, they provide a rigorous foundation for theorems and lemmas. In the sciences, proofs underpin the development of scientific theories and laws of nature.

History of Proof

The concept of proof has evolved over centuries, with significant contributions from ancient Greek mathematicians such as Euclid and Archimedes. The development of formal logic in the 19th and 20th centuries further refined the methods and standards of proof.

Related Concepts

• Axiom
• Theorem
• Lemma
• Corollary
• Hypothesis
• Scientific method
• Formal system
• Logical argument

See Also

• Mathematics
• Logic
• Scientific theory
• Number theory
• Combinatorics
• Euclid
• Archimedes