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HOMEWORK 5
Question 1. Prove by induction that for all n N,
n
i3 =
i=1
n(n + 1) 2
2
.
Question 2. Consider the sequence dened recursively by a1 = 2 an+1 = 7an + 9n + 5n Prove by induction that for all n 1, an = 9n 5n . 2
Question 3. Consider the sequence
HOMEWORK 6
Question 1. Prove that if A, B and C are any sets, then A (B C ) = (A B ) (A C ). Question 2. (a) Prove that if A, B , C and D are any sets, then (A B ) (C D) (A C ) (B D). (b) Give an example of sets A, B , C and D such that (A B ) (C D) = (A
Notations: we write Z+ = cfw_0, 1, 2, . . ., N = cfw_1, 2, 3, . . ., 2N := cfw_2n; n N = cfw_2, 4, 6, . . ., and 2N 1 := cfw_2n 1; n N = cfw_1, 3, 5, . . .. Properties In the following, : S S S is a binary operation on a nonempty set S . We write a b inst
Stylistic requirements vary from circumstance to circumstance and from teacher to teacher. What may be normal usage in other circumstances or in more advanced classes, may not be acceptable in a class whose primary focus is learning syntax and good style.