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'techniques in mathematical proofs'
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| Title of object: |
techniques in mathematical proofs |
| Canonical Name: |
TechniquesInMathematicalProofs |
| Type: |
Topic |
| Created on: |
2004-10-24 20:26:52 |
| Modified on: |
2004-10-24 21:57:09 |
| Classification: |
msc:00A35 |
| Defines: |
existential proof, constructive proof |
Revision comment (for changes between this and next version):
| fix a typo ("explicitely" changed to "explicitly") |
Preamble:
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Content:
\PMlinkescapeword{qed}
The following example (from ring theory) illustrates the one aspect of proofs in mathematics: proving the existence of certain mathematical objects or properties.
\par
\textbf{Statement:} Let $R$ be a ring such that $1-ab$ is right invertible, with $a,b\in R$. Then $1-ba$ is right invertible.
\par
\textbf{Existential proof:}
Since $1-ab\in R$ is right invertible, $(1-ab)R=R$. Now, $$(1-ba)R\supseteq(1-ba)bR=b(1-ab)R=bR.$$ So $$(ba)R=b(aR)\subseteq bR\subseteq\ (1-ba)R,$$ and consequently, $$R=(1-ba)R+(ba)R\subseteq (1-ba)R,$$ showing that $1\in(1-ba)R$. \qed
\par
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 $1-ba$.
\par
\textbf{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, $$(1-ba)bc=bc-babc=bc-b(c-1)=b.$$ 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 that
a right inverse of $1-ba$ exists by explicitely constructing one. \qed
\par
Many other techniques are used in proving mathematical statements. Proof by mathematical induction, proof by contradiction, proof by exhaustion, and proof by similarity are just some of the major techniques.
\par
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! |
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