(2) Then x + y = 2m +2n (by substitution). University. Proof. Discrete Mathematics - Lecture 1.8 Proof Methods and Strategy. Proof m = a2 and n = b2 for some integers a and b Then m + n + 2√ (mn) = a2 + b2 + 2ab = (a + b)2 So m + n + 2√ (mn) is a perfect square. Theorem 1. The argument is valid so the conclusion must be true if the premises are true. Assume that a and b are consecutive integers. Outline •What is a Proof ? It contains sequence of statements, the last being the conclusion which follows from the previous statements. Thus P(1) is true. ♦ Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed. 6. 7. Let P(n) represent " 2n − 1 is odd": (i) For n = 1, 2n − 1 = 2 (1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Course Discrete Mathematics (MATH 3336 ) Academic year.

Discrete Mathematics - Lecture 1.8 Proof Methods and Strategy. Proof: (1) There exist numbers m and n such that x = 2m and y = 2n (by def of “even”). Assume that P is true. First and foremost, the proof is an argument. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof.

2016/2017 What is a Proof ? = 2(m + n) (by left distrib) which is even, by the defn of evenness. Because a and b are Proving an Implication Claim: If r is irrational, then √r is irrational. This Lecture • Direct proof • Contrapositive • Proof by contradiction • Proof by cases. Discrete Mathematics Lecture 4 Proofs: Methods and Strategies 1 . Use P to show that Q must be true.

•A proof is a valid argument that establishes the truth of a theorem (as the conclusion) •Methods of Proving •Common Mistakes in Proofs •Strategies : How to Find a Proof ? Discrete Mathematics - Lecture 1.1 Propositional Logic Discrete Mathematics - Lecture 1.2 Applications of Propositional Logic Discrete Mathematics - Lecture 1.3 Propositional Equivalences Discrete Mathematics - Lecture 1.5 Nested Quantifiers Discrete Mathematics - Lecture 2.1 Sets Discrete Mathematics - Lecture 2.4 Sequences and Summations 2 . There are only two steps to a direct proof (the second step is, of course, the tricky part): 1. 2. the simplest and easiest method of proof available to us. (ii) For any n, if 2n − 1 is odd ( P(n) ), then (2n − 1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. If a and b are consecutive integers, then the sum a+ b is odd. University of Houston.

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