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techniques in mathematical proofs (Feature)

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

Statement: Let $ R$ be a ring such that $ 1-ab$ is right invertible, with $ a,b\in R$. 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 object or property 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 object or property in question.

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

$\displaystyle (1-ba)R\supseteq(1-ba)bR=b(1-ab)R=bR.$
So
$\displaystyle (ba)R=b(aR)\subseteq bR\subseteq\ (1-ba)R,$
and consequently,
$\displaystyle R=(1-ba)R+(ba)R\subseteq (1-ba)R,$
showing that $ 1\in(1-ba)R$.

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

Constructive proof: Since $ 1-ab\in R$ is right invertible, let $ c\in R$ 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,

$\displaystyle (1-ba)bc=bc-babc=bc-b(c-1)=b.$
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
$\displaystyle ba=(1-ba)bca.$
Then, negate both terms and add 1, to get
$\displaystyle 1-ba=1-(1-ba)bca.$
Finally, rearranging the terms and we have
$\displaystyle 1=(1-ba)+(1-ba)bca=(1-ba)(1+bca),$
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 contradiction, proof by contrapositive, and proof by exhaustion are just some of the major techniques (a simple example of the last type 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!



Anyone with an account can edit this entry. Please help improve it!

"techniques in mathematical proofs" is owned by CWoo. [ full author list (5) ]
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See Also: irrational to an irrational power can be rational, existential theorem, I-AB is invertible if and only if I-BA is invertible

Also defines:  existential proof, existence proof, constructive proof

Attachments:
an example of mathematical induction (Example) by CWoo
probabilistic proof (Example) by Algeboy
needle-in-the-haystack (Definition) by Algeboy
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Cross-references: contrapositive, proof by contradiction, induction, sides, term, equation, right inverse, right invertible, theory, ring
There are 8 references to this entry.

This is version 15 of techniques in mathematical proofs, born on 2004-10-24, modified 2007-06-13.
Object id is 6415, canonical name is TechniquesInMathematicalProofs.
Accessed 5773 times total.

Classification:
AMS MSC: 00A35 (General :: General and miscellaneous specific topics :: Methodology of mathematics, didactics)
 03F07 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: Structure of proofs)
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