1 Inference Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong

2 e.g.1 (Page 7) Property p  q is equivalent to  p  q pq p  q TTT TFF FTT FFT  p q p  q FTT FFF TTT TFT

3 e.g.2 (Page 17) Prove that: Let n be a positive integer. If n 2 > 100, then n > 10. Consider n  10 n. n  n. 10 (since n is a positive integer) n 2  n. 10  (since n  10) = 100 We conclude that n 2  100 (i.e., n 2 is not greater than 100.) We want to prove by contraposition. That is, “ If “ not (n > 10) ”, then “ not (n 2 > 100) ” “ If n  10, then n 2  100 ”

4 e.g.3 (Page 20) Illustration of “Proof by Contradiction” We are going to prove that a claim C is correct Proof by Contradiction: Suppose “ NOT C ” ….…. Derive some results, which may contradict to 1. “ NOT C ”, OR 2. some facts e.g., we derived that C is true finally e.g., we derived that “1 = 4”

5 e.g.4 (Page 27) E.g., m = 10 n = 21 Consider term “ m 2 ” and term “ 5n 2 ” We want to analyze the number of prime factors of m 2 (and 5n 2 ). m = 2. 5 n = 3. 7 Consider m 2 = (2. 5) 2 = Consider 5n 2 = 5. (3. 7) 2 = There are 4 prime factors (i.e., an even number of prime factors). There are 5 prime factors (i.e., an odd number of prime factors).