CIS 1910 Lecture Notes - Lecture 7: List Of Theorems, Luiza, Logical Reasoning

• School: University of Guelph
• Department: Computing and Information Science
• Course: CIS 1910
• Professor: Luiza Antonie

Document Summary

The average of two real numbers is less than or equal to at least one of the two numbers. If x r and y r, then x + y. Proofs by exhaustion: if the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually, for every positive integer n less than 3, (n + 1)2 3n. Proof by exhaustion: (1 + 1)2 31 (2 + 1)2 32. Every month has 30 or 31 days. Theorem: if n is an odd integer, then n2 is an odd integer. P (n) : n is an odd integer n = 2k + 1, k z. = 4 k2 + 4 k + 1. = 2 (2 k2 + 2 k) + 1. We can conclude that n2 is odd. Theorem: if x is a real number, and x 3, then 12 7x + x2 0. Let x r. xp (x) = q(x)