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Strategies for Proofs. 01/24/13. Landscape with House and Ploughman Van Gogh. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois. Goals of this lecture. Practice with proofs Become familiar with various strategies for proofs.

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01/24/13


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    1. Strategies for Proofs 01/24/13 Landscape with House and Ploughman Van Gogh Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois

    2. Goals of this lecture • Practice with proofs • Become familiar with various strategies for proofs

    3. Review: proving universal statements Claim: For any integer , if is odd, then is also odd. Definition: integer is odd iff for some integer

    4. Proving existential statements Claim: There exists a real number x, such that overhead

    5. Disproving existential statements Claim to disprove: There exists a real , In general, overhead

    6. Disproving universal statements Claim to disprove: For all real , In general, overhead

    7. Proof by cases Claim: For every real x, if , then

    8. Proof by cases Claim: For every real x, if , then

    9. Rephrasing claims Claim: There is no integer , such that is odd and is even.

    10. Proof by contrapositive Claim: For all integers and ,

    11. Proof strategies • Does this proof require showing that the claim holds for all cases or just an example? • Show all cases: prove universal, disprove existential • Example: disprove universal, prove existential • Can you figure a straightforward solution? • If so, sketch it and then write it out clearly, and you’re done • If not, try to find an equivalent form that is easier • Divide into subcases that combine to account for all cases • OR in hypothesis is a hint that this may be a good idea • Try the contrapositive • OR in conclusion is a hint that this may be a good idea • More generally rephrase the claim: convert to propositional logic and manipulate into something easier to solve

    12. More proof examples Claim: For integers and , if is even or is even, then is even. Definition: integer is even iff for some integer

    13. More proof examples Claim: For all integers , if is even, then is odd.

    14. Another proof Claim: For any real , if is rational, then is rational. Definition: real is rational ifffor some integers and , with . overhead

    15. More proof examples Claim: For all integers , if is odd, then or for some integer . (Note, this requires knowing a little about modular arithmetic.)