Exercise 1.2.7. Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; that is, f(A) = {f(x): x = A}. (a) Let f(x) = x2. If A = [0,2] (the closed interval {x = R: 0≤ x ≤ 2}) and B = [1,4], find f(A) and f(B). Does f(An B) = f(A)nf(B) in this case? Does f(AUB) = f(A) U f(B)? وفاز (b) Find two sets A and B for which f(An B) f(A) nƒ(B). (c) Show that, for an arbitrary function g: RR, it is always true that g(An B) C g(A) ng(B) for all sets A, B C R. da lo w bhi 8.S.I (d) Form and prove a conjecture about the relationship between g(AUB) and g(A) Ug(B) for an arbitrary function g. to soc

Exercise 1.2.5 (De Morgan's Laws). Let A and B be subsets of R. (a) If x = (An B), explain why x € AU Bc. This shows that (An B)c C AC U Bc.

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10- Prove that a.b= la+bl?- la-bl? /4

Let 6. 1] M 2 9. Find formulas for the entries of M", where n is a positive integer. M" ||