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单词 TechniquesInMathematicalProofs
释义

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,

(1-ba)R(1-ba)bR=b(1-ab)R=bR.

So

(ba)R=b(aR)bR(1-ba)R,

and consequently,

R=(1-ba)R+(ba)R(1-ba)R,

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,

(1-ba)bc=bc-babc=bc-b(c-1)=b.

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

ba=(1-ba)bca.

Then, negate both terms and add 1, to get

1-ba=1-(1-ba)bca.

Finally, rearranging the terms and we have

1=(1-ba)+(1-ba)bca=(1-ba)(1+bca),

showing thata 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!

Titletechniques in mathematical proofs
Canonical nameTechniquesInMathematicalProofs
Date of creation2013-03-22 14:46:15
Last modified on2013-03-22 14:46:15
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id18
AuthorCWoo (3771)
Entry typeFeature
Classificationmsc 00A35
Classificationmsc 03F07
Related topicIrrationalToAnIrrationalPowerCanBeRational
Related topicExistentialTheorem
Related topicIABIsInvertibleIfAndOnlyIfIBAIsInvertible
Definesexistential proof
Definesexistence proof
Definesconstructive proof
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