Proof
Pronunciation: /pruf/ Explain
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.
References
- Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs. 3rd edition. Academic Press. August 15, 2005. Buy the book
More Information
- Disproof. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 1/11/2010. http://www.allmathwords.org/en/d/disproof.html.
Cite this article as:
McAdams, David E. Proof. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/proof.html.