Lecture 9 Enoch Cheung September 30 2013 1 What is sn 12 22 32 n2 in summation notation Prove that sn n n 1 2n 1 6 In summation notation sn n X i2 i 1 Base case s1 1 1 2 3 6 1 Inductive case Suppose sn sn 1 n 1 X n n 1 2n 1 6 we want to show sn 1 n 1 n 2 2n 3 6 i2 i 1 n X i2 n 1 2 i 1 n n 1 2n 1 n 1 2 6 n n 1 2n 1 6 n 1 2 6 n 1 n 2n 1 6 n 1 6 n 1 2n2 7n 6 6 n 1 n 2 2n 3 6 by the inductive hypothesis 2 Let t1 t2 t3 1 and tn tn 1 tn 2 tn 3 Show that tn 2n for all n N Base cases t1 1 2 t2 1 4 22 t3 1 8 23 Inductive case Suppose tk 2k for all k n We wish to show tn 2n tn tn 1 tn 2 tn 3 2n 1 2n 2 2n 3 1 1 1 2n 2n 2n 2 4 8 1 1 1 2n 2 4 8 7 2n 8 2n 3 Let Ai i N and Bi i N be two indexed families of sets such that for all i N Ai Bi Show that Ai Bi i N i N 1 Is it true always true that Ai i N Let x S i N Bi i N Ai then x Ak for some k so x Bk 1 S i N Bi It is not true Conisder Ai i and Bi i 1 Certainly AiS Bi However consider x then x Bk k 1 for some k so x Ak 1 k 1 so x i N S i N Bi 4 Prove or disprove P A A 6 P A for all sets A 6 This is false Let A then P A 5 Let A B and C D Prove or disprove A C B D Suppose x y A C then x A B and y C D so x y B D 6 Prove or disprove A B C D A C B D for all A B C D Let A 2 B 2 C 1 D 2 then A B C D 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 However A C B D 2 2