Downloaded 14 times
More Related Content
Similar to Unit 1 introduction to proofs
Unit 1 introduction to proofs
- 1.
- 2. terminology • Theorem/facts/results: • Astatement 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 • Aless 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 provingtheorems • Direct proof • Proofs by contraposition • Vacuous and trivial proofs • Proofs of equivalence • counterexamples
- 5. Direct proof • Adirect 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 evenand 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 • Twointegers 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 adirect proof of the theorem “If n is an odd integer, then n2 is odd.”
- 9. example • Give adirect proof that if m and n are both perfect squares, then nm is also a perfect square.
- 10.
- 11.