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Proofs • A theorem is a mathematical statement that can be shown to be true. • An axiom or postulate is an assumption accepted without proof. • A proof is a sequence of statements forming an argument that shows that a theorem is true. The premises of the argument are axioms and previously proved theorems. • A lemma is a short theorem used in the proof of another theorem. • A corollary is a theorem that follows directly from another theorem. • A conjecture is a mathematical statement whose truth value is still unknown. Once proved (if it is indeed true), it becomes a theorem. • A fallacy is an incorrect reasoning. • Common fallacies: – Fallacy of affirming the conclusion – Fallacy of denying the hypothesis – Begging the question (or circular reasoning) • Methods of proofs... Type of Proof Direct proof Indirect proof (proof by contraposition) Proof by contradiction Proof by cases Proof of an equivalence Vacuous proof Trivial proof Proposition to Prove Strategy Examples of proofs (by type) Theorems to be proved in class, and required definitions Definition 1 An integer n is called odd if n = 2k + 1 for some integer k, and is called even if n = 2m for some integer m. Direct proof Theorem 2 If n is an odd integer, then n2 is an odd integer. Indirect proof (proof by contraposition) Theorem 3 If 5n + 4 is an odd integer, then n is an odd integer. Definition 4 Let m and n be positive integers. If n = km for some positive integer k, then we say that • n is a multiple of m; • m is a divisor of n; • m divides n; and we write m|n (read “m divides n”). Definition 5 A real number r is called rational if r = pq for some integers p and q with q ̸= 0. A real number that is not rational is called irrational. Proof by contradiction √ Theorem 6 2 is irrational. Proof by cases Theorem 7 On the Island of Knights and Knaves you meet three people, A, B, and C. If A says: “All of us are knaves,” and B says: “Exactly one of us is a knave,” then C must be a knight. Proof of equivalence Theorem 8 Let m and n be positive integers. Then m = n if and only if m|n and n|m. Vacuous proof Theorem 9 If 0 > 1, then √ 2 is a rational number. Trivial proof Theorem 10 If 0 < 1, then √ 4 is a rational number. More examples of proofs (mixed types) Theorem 11 The equation x3 + x + 1 = 0 has no rational roots. Theorem 12 Let a and b be real numbers. The following statements are equivalent: 1. a < b 2. a+b 2 >a 3. a+b 2 <b Additional exercises (and to be used in the proof of Theorem 11) Choose the most suitable proof type to prove the following. Lemma 13 Let n be an integer. The following are equivalent: 1. n is even. 2. n2 is even. 3. n3 is even. Lemma 14 Let m and n be integers. Then mn is odd if and only if m and n are both odd. Lemma 15 Let m and n be integers. Then m + n is odd if and only if exactly one of m and n is odd.