Problem 5. Let S be a subset of a universal set U. The characteristic function fs of S is the function from U to the set {0, 1} such that fs(x) = 1 if x belongs to S and fs = 0 if x does not belong to S. Let A and B be sets. Show that for all x € U, Note: We can prove all of these identities by showing that the left-hand side is equal to the right-hand side for all possible values of x. In each instance, there are four cases to consider, depending on whether x is in A and/or B. 5(a) fanB(x) = fA(x) · fB(x) 5(b) fAUB(x) = fA(x)+ fB(x) – fA(x) · fB(x) %3D 5(c) fA©B(x) = fA(x)+ fB(x) – 2fA(x)fB(x)

1. Suppose f: A → B is an injective function. Prove that f-1(f(C)) = C for all CCA. 2. Suppose A, B are sets, f : A → B is a function, and D₁, D₂ C B. In lectures we noted that the inverse image is well behaved with respect to set operations: f-¹ (D₁ D₂) = f¯¹ (D₁) n f¯¹ (D₂), f-¹ (D₁U D₂) = f¯¹ (D₁) Uf-¹ (D₂), and f¯-¹ (D₁ A D₂) = f¯¹ (D₁) A f¯¹(D₂). The image is not so well behaved. Yet one of the above three identities continues to hold with f replacing f-1 (and with D₁, D₂ replaced by subsets of A). Which is it? Can you prove this identity?

2. If f(2) = e*, describe the images under f(z) of horizontal and vertical lines, i.e. what are the sets f(a + it) and f(t + ib), where a and b are constants and t runs through all real numbers?

3. Let S {1,2, 3, 4} and define functions f,g : S → S by f = {(1,3), (2,2), (3, 4), (4, 1)} and g = {(1, 4), (2, 3), (3, 1), (4, 2)}. Find (a) g-1 o f og (b) f oglog (c) f-1 og –1 -1ofog