Unit 1 introduction to proofs

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This document introduces various concepts and methods related to mathematical proofs. It defines key terminology like theorems, propositions, lemmas, corollaries, and conjectures. It also describes different types of proofs like direct proofs, proofs by contraposition, and proofs of equivalence. Examples are provided to illustrate direct proofs for statements about odd integers and perfect squares.

terminology
• Theorem/facts/results:
• A statement of some importance that can be shown to be true
• Propositions:
• Less important statements which can be shown to be true
• Proof
• A valid argument that establishes the truth of the theorem

• Lemma
• A less important theorem that is helpful in the proof of other results
• Corollary
• Theorem that can be established directly from a theorem that has been
proved
• Conjecture
• A statement that is being proposed to be true statement on basis of
partial evidence, intuition of an expert

Methods of proving theorems
• Direct proof
• Proofs by contraposition
• Vacuous and trivial proofs
• Proofs of equivalence
• counterexamples

Direct proof
• A direct proof of a conditional statement p → q is
constructed when the first step is the assumption that p is
true; subsequent steps are constructed u
• A direct proof shows that a conditional statement p → q is
true by showing that if p is true, then q must also be true, so
that the combination p true and q false never occurs. Using
rules of inference, with the final step showing that q must
also be true

Definition of even and odd integers
• The integer n is even if there exists an integer k such that n
= 2k, and n is odd if there exists an integer k such that n
= 2k + 1

Same parity
• Two integers have the same parity when both are even or
both are odd; they have opposite parity when one is even
and the other is odd.

example
• Give a direct proof of the theorem “If n is an odd
integer, then n2 is odd.”

example
• Give a direct proof that if m and n are both perfect
squares, then nm is also a perfect square.

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Unit 1 introduction to proofs

1. Introduction to proofs

2. terminology • Theorem/facts/results: • A statement of some importance that can be shown to be true • Propositions: • Less important statements which can be shown to be true • Proof • A valid argument that establishes the truth of the theorem

3. • Lemma • A less important theorem that is helpful in the proof of other results • Corollary • Theorem that can be established directly from a theorem that has been proved • Conjecture • A statement that is being proposed to be true statement on basis of partial evidence, intuition of an expert

4. Methods of proving theorems • Direct proof • Proofs by contraposition • Vacuous and trivial proofs • Proofs of equivalence • counterexamples

5. Direct proof • A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed u • A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. Using rules of inference, with the final step showing that q must also be true

6. Definition of even and odd integers • The integer n is even if there exists an integer k such that n = 2k, and n is odd if there exists an integer k such that n = 2k + 1

7. Same parity • Two integers have the same parity when both are even or both are odd; they have opposite parity when one is even and the other is odd.

8. example • Give a direct proof of the theorem “If n is an odd integer, then n2 is odd.”

9. example • Give a direct proof that if m and n are both perfect squares, then nm is also a perfect square.

10. Proof by contraposition

11. Mistakes in proofs

Unit 1 introduction to proofs

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