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Strategies for Proofs Examples Strategies for Proofs G. Carl Evans University of Illinois Summer 2013 Strategies for Proofs Strategies for Proofs Examples Today Practice with proofs Become familiar with various strategies for proofs Strategies for Proofs Strategies for Proofs Examples Review: proving universal statements Claim: For any integer a, if a is odd, then a2 is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Suppose a ∈ Z is odd. Then by definition of odd, a = 2m + 1 for some m ∈ Z. So a2 = (2m + 1)2 = 4m2 + 4m + 1 = 2(2m2 + 2m) + 1 2m2 + 2m is an integer since m ∈ Z. So a2 is odd by definition QED. Strategies for Proofs Strategies for Proofs Examples Proving Existential Statements Claim: There exists a real number x, such that |x 3 | < x 2 Let x = 12 . Then |x 3 | = 81 and x 2 = 1/4. Since an x where |x 3 | < x 2 QED. Strategies for Proofs 1 8 < 1 4 there exists Strategies for Proofs Examples Disproving Existential Statements Claim to disprove: There exists a real x, x 2 − 2x + 1 < 0 Strategies for Proofs Strategies for Proofs Examples Disproving Existential Statements Claim to disprove: There exists a real x, x 2 − 2x + 1 < 0 In general, ¬(∃x, P(x)) ≡ ∀x, ¬P(x) ¬(∃x ∈ R, x 2 − 2x + 1 < 0) ∀x ∈ R, ¬(x 2 = 2x + 1 < 0) ∀x ∈ R, x − 2x + 1 ≥ 0 Let x be a real number. x 2 − 2x + 1 = (x − 1)2 . (x − 1)2 ≥ 0 since x − 1 is a real number and the square of a real number is non-negative QED. Strategies for Proofs Strategies for Proofs Examples Disproving Universal Statements Claim to disprove: For all real x, (x + 1)2 > 0 Strategies for Proofs Strategies for Proofs Examples Disproving Universal Statements Claim to disprove: For all real x, (x + 1)2 > 0 In general, ¬(∀x, P(x)) ≡ ∃x, ¬P(x) ¬(∀x ∈ R, (x + 1) > 0) ∃x ∈ R, (x + 1)2 ≤ 0) If x = 1, (x + 1)2 = 0. So since 1 is a real there then exists a real that proves the negation of the claim and thus the original claim is disproved QED. Strategies for Proofs Strategies for Proofs Examples Proof By Cases Claim: for every real x, if |x + 7| > 8, then |x| > 1 Suppose x ∈ R and |x + 7| > 8. Then there are two cases. If x + 7 > 8, then x > 1, so |x| > 1. If x + 7 < −8, then x < −15, so |x| > 1. The conclusion holds for both cases QED. Strategies for Proofs Strategies for Proofs Examples Rephrasing Claims Claim: There is no integer k, such that k is odd and k 2 is even. ¬(∃k ∈ Z, odd(k) ∧ even(k 2 ) ∀k ∈ Z, ¬(odd(k) ∧ even(k 2 )) ∀k ∈ Z, ¬odd(k) ∨ ¬even(k 2 ) ∀k ∈ Z, even(k) ∨ odd(k 2 ) At this point there is a proof by cases. Strategies for Proofs Strategies for Proofs Examples Proof By Contrapositive Claim: For all integers a and b, (a + b ≥ 15) → (a ≥ ∨b ≥ 8) Contrapositive: ∀a, b ∈ Z¬(a ≥ 8 ∨ b ≥ 8) → ¬(a + b ≥ 15) ∀a, b ∈ Z, a < 8 ∧ b < 8 → a + b < 15 To prove the claim we will prove the contrapositive, ∀a, b ∈ Z, a < 8 ∧ b < 8 → a + b < 15. Suppose a ∈ Z, a < 8 and a ∈ Z, b < 8. Then a ≤ 7 and b ≤ 7. So a + b ≤ 14 < 15 QED. Strategies for Proofs Strategies for Proofs Examples 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 youre 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 Strategies for Proofs Strategies for Proofs Examples Claim: For integers j and k, if j is even or k is even, then jk is even. Definition: integer a is even iff a = 2m for some integer m Strategies for Proofs Strategies for Proofs Examples Claim: For all integers k, if 3k + 5 is even, then k is odd. Strategies for Proofs Strategies for Proofs Examples Disprove: For all real k, if k is rational, then k3 k Strategies for Proofs is rational. Strategies for Proofs Examples Alternate proof, by cases Claim: For any real k, if k is rational, then k 2 is rational. Definition: real k is rational iff k = m n for some integers m and n with n 6= 0 Strategies for Proofs Strategies for Proofs Examples Claim: For all integers x, if x is odd, then x = 4k + 1 or x = 4k − 1 for some integer k. Strategies for Proofs