Assignment:
Q1. Let X = {1, 2, 3} and Y = {-1, -2, -3}. Define the new set X o Y = {z: z = xy for x is an element of X and y is an element of Y}. This new set X o Y is obtained by taking products of pairs of element one from X and the other from Y. Is Y a subset of X o Y? If yes, tell me why -1 is an element of X o y, -2 is an element of X o Y and -3 is an element of X o Y?
Q2. Suppose that there are three men and three women. Let us denote the set of men by M = {m1, m2, m3} and set of women by W = {w1, w2, w3}. The ranking matrix for these set of men and women is given by
w1 w2 w3
m1 3,3 1,2 2,1
m2 1,2 2,1 3,2
m3 2,1 1,3 3,3
Is the following marriage configuration a stable marriage (that is, is it divorce proof): m1 married to w2; m2 married to w1; m3 married to w3? If you answer yes, show that no man-woman pair want to break up the marriage they are in. If you answered no, find a man-woman pair that is willing to break up their current marriage with each other.
Q3. Let f(x) = sqrt(x ln x). Derive the expression for the first derivative of f, f'(x). [Recall: the derivatie of ln(x) is (1/x)].
Q4. Let A = {x is an element of R: x^2 - 4 ≤ 0}. Express teh set A as a closed interval [a,b] for some a < b. Give an explanation as to why you think your answer correctly identifies all elements of A with your proposed closed interval.
Provide complete and step by step solution for the question and show calculations and use formulas.