07au366templates - PROOF TEMPLATES 1 Basic form of a proof...

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PROOF TEMPLATES 1. Basic form of a proof of an existential statement using the method of constructive proof. 2. Proving an existential statement without a domain using the method of constructive proof. 3. Basic form of proofs using the method of direct proof. 4. Proofs using the method of direct proof in other situations. 5. Basic form of proofs using the method of division into cases. 6. Basic form of proofs using the method of argument by contradiction. 7. Basic form of proofs using the method of argument by contraposition. 8. The form of proofs using the principle of mathematical induction. 9. The form of proofs using the principle of strong mathematical induction. 1
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BASIC FORM OF A PROOF OF AN EXISTENTIAL STATEMENT USING THE METHOD OF CONSTRUCTIVE PROOF Suppose you are trying to prove a statement which has the following logical form x D s.t. P ( x ) Before beginning your proof, take out some scratch paper and find some particular a in D with the property that P ( a ) is true. You can now give a dirct proof of the statement which looks like (instructions and comments are written in italics): Proof. Let x = a . write out a proof that x D here * * * write out a proof that P ( x ) here * * * We have established that there exists an x in D such that P ( x ). QED You might prefer the following alternative. Proof. write out a proof that a D here * * * write out a proof that P ( a ) here * * * Since a D and P ( a ), there exists an x in D such that P ( x ). QED 2
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PROVING AN EXISTENTIAL STATEMENT WITHOUT A DOMAIN USING THE METHOD OF CONSTRUCTIVE PROOF Suppose the statement you are trying to prove is like that on the previous page but is missing the domain D : x s.t. P ( x ) As on the previous template, before beginning your proof, take out some scratch paper and find some particular a with the property that P ( a ) is true. You can now give a dirct proof of the statement which looks like (instructions and comments are written in italics): Proof. Let x = a . write out a proof that P ( x ) here * * * We have established that there exists an x such that P ( x ). QED You might prefer the following alternative. Proof. write out a proof that P ( a ) here * * * Since P ( a ), there exists an x such that P ( x ). QED 3
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BASIC FORM OF PROOFS USING THE METHOD OF DIRECT PROOF Suppose you are trying to prove a statement which has the following logical form x D, if P ( x ) then Q ( x ) Here is what a direct proof of that statement should look like. Instructions and comments are written in italics.
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