techniques in mathematical proofs

The following example (from ring theory) illustrates the one aspect of proofs in mathematics: proving the existence of certain mathematical .

Statement: Let R be a ring such that 1-ab is right invertible, with a,bR. Then 1-ba is right invertible.

This statement will be proven here using two methods. The first method is called an existential proof (also known as an existence proof), in which one only seeks to prove that the mathematical in question exists, not to show how to obtain it. The second method is called a constructive proof, in which one actually shows how to obtain the mathematical in question.

Existential proof: Since 1-abR is right invertible, (1-ab)R=R. Now,




and consequently,


showing that 1(1-ba)R.       

Notice, we merely demonstrated the existence of a right inverseMathworldPlanetmath of 1-ba without actually finding such an . The next proof in fact finds a right inverse of 1-ba.

Constructive proof: Since 1-abR is right invertible, let cR be a right inverse so that 1=(1-ab)c. We seek to construct a right inverse of 1-ba in terms of a,b, and c. Rewriting the equation, we have abc=c-1. Then,


We have just expressed b in terms of 1-ba. Next, multiply a on the right to each term on both sides of the equation, to get


Then, negate both terms and add 1, to get


Finally, rearranging the terms and we have


showing that a right inverse of 1-ba exists by explicitly constructing one.       

Many other techniques are used in proving mathematical statements. Proof by mathematical induction, proof by contradictionMathworldPlanetmathPlanetmath, proof by contrapositive, and proof by exhaustion are just some of the major techniques (a is in the entry “irrational to an irrational power can be rational”).

As this entry is still in its very rough form, PM users are welcome and encouraged to refine and provide additional techniques with interesting and illustrative examples!

Title techniques in mathematical proofs
Canonical name TechniquesInMathematicalProofs
Date of creation 2013-03-22 14:46:15
Last modified on 2013-03-22 14:46:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 18
Author CWoo (3771)
Entry type Feature
Classification msc 00A35
Classification msc 03F07
Related topic IrrationalToAnIrrationalPowerCanBeRational
Related topic ExistentialTheorem
Related topic IABIsInvertibleIfAndOnlyIfIBAIsInvertible
Defines existential proof
Defines existence proof
Defines constructive proof