Download Class Notes (Jan.30)

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
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Type of Proof
Proposition to Prove
Strategy
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Proof by cases
Proof of an
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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
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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
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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
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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.