07au366templates

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

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern