Download Proofs • A theorem is a mathematical statement that can be shown to

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.