Pronunciation: /pruf/ ?

A proof is a logical argument that shows that a claim is always true.[1] A mathematical proof must show that a claim is true in all cases, without any exceptions. The opposite of a proof is a disproof is the opposite of a proof: an argument that shows a claim is always false. A claim that has not been proved always true and has not been proved always false, but is generally believed to be true is called a conjecture.

Types of Proofs

The are many types of proofs. Some of the types that are commonly used are:
  • Direct proof: A direct proof is built on axioms, definitions and previously proved theorems.
  • Proof by induction: An inductive proof is used to proof claims about infinite sets. An inductive proof shows that if a claim is true for the first case and for an arbitrary case, it is always true for the next case after the arbitrary case.
  • Proof by transposition: A proof by transposition shows that the contrapositive of a statement is true. Since the contrapositive of a statement is always true if the original statement is true, the statement is taken to be true.
  • Proof by contradiction: A proof by contradiction starts with a claim. The assumption is made that the claim is false. The proof then shows that if the claim is false a contradiction is reached. The claim must then be true.
  • Proof by exhaustion: In proof by exhaustion, a claims is divided into a number of cases, and each of the cases is individually proved.
  • Proof by construction: In proof by construction, a concrete example is 'constructed' with a property that shows that something with that property exists. A proof by construction can also be called a proof by example.
  • Flow proof: A flow proof is a proof where each statement and its justification are placed in a box and arrows show the logical flow from one box to another.


  1. proof. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs, 3rd edition. Academic Press, June 3, 2005.

More Information

  • Disproof. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-01-11.
  • Mastering the Formal Geometry Proof (video). Wiley. 2010-01-23.

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Proof. 2010-01-11. All Math Words Encyclopedia. Life is a Story Problem LLC.

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2010-01-11: Initial version (McAdams, David.)

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