Let Z be the set of all integers and let {n € Z]n = 4k, for some integer k} {n € Z]n = 4k + 1, for some integer k} A2 {n €Z]n = 4k + 2, for some integer k} and Az = {n € Z |n = 4k + 3, for some integer k}. s{Ao•Ag Az, Ag a partition of Z? Why or why not? O Yes. Z = A, U A1 U A, U A3, and A, n Az = Ø, A, N A2 = Ø, A, N Az = Ø, A N Az = Ø, A, N A3 = Ø, A, N Az = Ø. O Yes. Z = A, nA, N A, N A3, and A, U A, = Ø, A, U A, = Ø, A, U A3 = Ø, A, U A2 = Ø, A, U A3 = 0, A2 U A, = Ø. O No. Z + A, U A, U A, U A, and A, NA, # Ø. O No. Z = A, n A, N but A, U A, # Ø. O No. Z = A, U A, UA, U A3, but A, N A, # Ø.

the suprema and infima of the following sets: (a) {n €N | n² < 10}; (b) { | m, n = N}; n m+n n (c) {2+1 ne N}; (d) { | m, n € N with m + n ≤ 10}. m Compute, without proof,

Pick an interesting and original positive integer k with10 ≤ k. Also pick an interesting and original positive integer n withn ≥ 30 and n > k.(a) How many permutations of size k of n objects are there? Explainhow you compute your answer.(b) How many combinations of size k of n objects are there? Explainhow you compute your answer.(c) Suppose a bin contains n balls numbered 1, 2, . . . , n and 3 areselected at random. What is the probability that the the threeballs are all numbered between 5 and k (inclusive of the numbers5 and k).

Define a digit to be an element of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 00 00 Show that Σ n=1 10" = Σ n n=1 b - if and only if one of the following two conditions holds: 10" Either (i) a = b_for every n or 12 (ii) there is some N&N U {0} such that a = b for 1 ≤ n ≤ N, 12 a = b₁ + 1, and N a = 0 and b = 9 for all n > N. 12 12