For each of the following pairs of functie ff and gg, circle one of the answers f \in o(g), f \in \Theta(g),fEo(g),f€0(g), or g \i o(f)gEo(f) f(n) = 2", g(n) = (;) Oƒ € o(g) Oƒ€ 0(g) %3D O ge o(f) i=log2 n 2i, g(n) = n f(n) = C Oƒ € o(g) %3D %3D Oƒe0(g) O ge o(f) f(n) = (2'º + 3n)², g(n) = n² %3D %3D Oƒ€ o(g) Oƒ€ 0(g) O ge o(f)
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- Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X BSuppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1What are the tightest bounds for the functions Tị and T2 (respectively from left to right) such that: T1 (n) = T1 () + cn Т, (п) — Т, (п - 4) + сп
- If A = {0, 1), B = {1, 2, 3), then (AUB) x B is equal to:Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + *1 and g(x) = 3x + 4. Then the composition of fog is %3DDetermine whether each of these functions is a bijection from ℝ to ℝ. If it is, write the inverse function. f(x) = 3|x| − 4
- Ql: The Collatz conjecture function is defined for a positive integer m as follows. (COO1) g(m) = 3m+1 if m is odd = m/2 if m is even =1 if m=1 The repeated application of the Collatz conjecture function, as follows: g(n), g(g(n)), g(g(g(n))), ... e.g. If m=17, the sequence is 1. g(17) = 52 2. g(52) = 26 3. g(26) = 13 4. g(13) = 40 5. g(40) = 20 6. g(20) = 10 7. g(10) = 5 8. g(5) = 16 9. g(16) = 8 10. g(8) = 4 11. g(4) = 2 12. g(2) = 1 Thus if m=17, apply the function 12 times in order to reach m=1. Use Recursive Function.6 For the following pairs of functions, first decide whether f(n) dominates g(n), or g(n) dominates f(n); then decide whether f = 0(g), or f = N(g), or f = 0(g), and briefly explains. i. f(n) = n²,g(n) = 1000n + 30 ii. f(n) = /n, g(n) = n} iii. f(n) = 10log2 n, g(n) = log10(n³) iv. f(n) = n100, g(n) = 1.2".Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.
- The function of four variables G (A, B, C, D) = Z (2, 3, 8, 9, 11, 14, 15) + F (6, 10) is given. Here F values are "Doesn't matter" (Indeterminate) values that can take any value. Write this function in a simplified form using the Karnaugh Map.Determine whether each of these functions is a bijection from ℝ to ℝ. If it is, write the inverse function. ?(?) = 3|?| − 4The 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…