Consider a function f : S→T defined by f(n) = n², where the domain is the set of consecutive integers S = {-7,..., -2, -1, 0, 1, 2, ..., 7} and the range is the set of consecutive squares T = {-64,..., -9,–4, −1, 0, 1, 4, 9, . . . ,64}. Which one of the following choices correct describes f? Oone-to-one, but NOT onto Oonto, but NOT one-to-one O one-to-one and onto Oneither one-to-one nor onto Onone of these

Let F be the set of all functions of the form f : N {0,1}. Also, let G be the set of all functions of the form f: {0, 1}* → {a, b}.Then: O (a) |F| = |G| = |N| O (b) |F| = |P(G)| 미 (c) 1지=1P(N)| □ (d) 지< IgI, but |F # I9| □ (e) |지= 19| %3D %3D Select all possible options that apply. e

Let i, j ∈ Za and w, x ∈ Zb with i, j different from each other and w, x different from each other. When a function f : Za → Zb is chosen at random, what is Pr[f(j) = x | f(i) = w]?

The question describes a function S(k) which is defined as the sum of the positive divisors of a positive integer k, minus k itself. The function S(1) is defined as 1, and for any positive integer k greater than 1, S(k) is calculated as S(k) = σ(k) - k, where σ(k) is the sum of all positive divisors of k. Some examples of S(k) are given: S(1) = 1 S(2) = 1 S(3) = 1 S(4) = 3 S(5) = 1 S(6) = 6 S(7) = 1 S(8) = 7 S(9) = 4 The question then introduces a recursive sequence a_n with the following rules: a_1 = 12 For n ≥ 2, a_n = S(a_(n-1)) Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence. Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific…