# Various Set Applications And Ranking Matrix

b) 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?

c) 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.

d) 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)].

e) 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.

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#### Solution Preview

b. First XY={-1,-2,-3,-4,-6,,-9}. Since all elements of Y belongs to XY, so yes YâŠ‚XY. Because -1=1*(-1) where 1 is from X and -1 is from Y, so

-1âŠ‚XY. Similarly, Because ...

#### Solution Summary

The solution gives detailed steps on some calculations in the set theory, determining whether the given rank matrix is stable. Also, a question about finding derivative using multiplication rule is shown and explained step by step.