Name: SSN: Grade:MA334B EXAM 2 November 2001I pledge my honor that I have abided by the Stevens Honor System.1 (12pts)• Let Σ = {a, b, c} and let f : Σ∗→ Σ∗be the function given by f(w) = awb. Is f 1-1, onto? Ifnot, explain.• Let g : N → N be the function given by g(n) = n3. Is f 1-1, onto? If not, explain.2 (12pts) Let X = {p, q, r, s}, Y = {t, u} and Z = {a, b, c}.• How many relations are there on the set X?• How many functions have domain X and codomain Z?• How many onto functions have domain Z and codomain Y ?3. (12pts)Show that if 17 points are randomly placed in a unit square, at least 2 of them will be no furtherthan12√2apart.4 (12pts) Let A = {x ∈ R | 0 < x < 2}.• Define: A set X is a countably infinite set.• Is A a countably infinite set? Prove or disprove.5 (16pts)Let the functions f, g : N → N be given by f(n) = n + 1 and g(n) = n3. Find• (g o f)(x)• f2(2)• f(A), where A = {1, 2}• g−1(B), where B = {1, 27}.6 (12pts) Let A = {1, 2, 3, 4}.• Define an equivalence relation R on A in which (2, 3) ∈ R and (1, 4) /∈ R.• What is [4] in your relation?7 (24pts)FindConsider the following relations on A. Are they reflexive, symmetric, antisymmetric or transitive?If they are, simply note this by putting the letter R, S, A or T next to the relation. If not, explainwhy not.1. A = {0, 1, 2, 3}. R1= {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 3), (3, 3)}.2. A = Z. R2: {(x, y) | x = y2}.3. A = Z. R3: {(x, y) | xy ≥