Mathematical proof

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== Mathematical Proof ==

A mathematical proof is a logical argument demonstrating the truth of a given statement based on previously established statements such as theorems, axioms, and lemmas. Proofs are essential in mathematics as they provide a rigorous foundation for mathematical knowledge.

Types of Proofs

There are several types of mathematical proofs, including:

• Direct proof: A method of proof that involves a straightforward chain of logical deductions from the given assumptions to the statement being proved.
• Indirect proof: Also known as proof by contradiction, this method assumes the negation of the statement to be proved and derives a contradiction.
• Proof by induction: A method used to prove statements about natural numbers. It involves proving a base case and an inductive step.
• Proof by construction: Demonstrates the existence of a mathematical object by explicitly constructing it.
• Proof by exhaustion: Involves checking all possible cases to prove a statement.

Structure of a Proof

A typical mathematical proof consists of the following components:

• Statement of the theorem: The proposition or theorem that is to be proved.
• Assumptions: The axioms, definitions, and previously established results that are used in the proof.
• Logical deductions: A sequence of logical steps that lead from the assumptions to the conclusion.
• Conclusion: The final statement that completes the proof.

Importance of Proofs

Mathematical proofs are crucial for several reasons:

• They ensure the correctness of mathematical statements.
• They provide insight into why a statement is true.
• They help in the discovery of new mathematics by revealing connections between different areas.

Famous Proofs

Some of the most famous mathematical proofs include:

• Euclid's proof of the infinitude of prime numbers
• Pythagorean theorem
• Fermat's Last Theorem
• Gödel's incompleteness theorems

Related Pages

• Theorem
• Axiom
• Lemma
• Corollary
• Mathematical logic
• Formal system
• Proof theory

See Also

• List of mathematical proofs
• Mathematical notation
• Mathematical rigor
• Mathematical induction
• Constructive proof

External Links

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