Discrete mathematical structures-mca vtu,1st semester cbcs scheme-module system-module 1

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