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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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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
Review: proving universal statements Claim: For any integer , if is odd, then is also odd. Definition: integer is odd iff for some integer
Proving existential statements Claim: There exists a real number x, such that overhead
Disproving existential statements Claim to disprove: There exists a real , In general, overhead
Disproving universal statements Claim to disprove: For all real , In general, overhead
Proof by cases Claim: For every real x, if , then
Proof by cases Claim: For every real x, if , then
Rephrasing claims Claim: There is no integer , such that is odd and is even.
Proof by contrapositive Claim: For all integers and ,
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
More proof examples Claim: For integers and , if is even or is even, then is even. Definition: integer is even iff for some integer
More proof examples Claim: For all integers , if is even, then is odd.
Another proof Claim: For any real , if is rational, then is rational. Definition: real is rational ifffor some integers and , with . overhead
More proof examples Claim: For all integers , if is odd, then or for some integer . (Note, this requires knowing a little about modular arithmetic.)