Mathematics Question

    . Let X be an arbitrary set and suppose that A, B and C are subsets of X. One of the following statements is true, and one is false. Give a careful proof of the statement that is true, and provide a specifific counterexample to the statement which is false. (This means that for the statement that is false, you should specify sets X, A, B and C, and explain why the statement is false for the sets you have specifified.) (a) (A B) ∩ C = (A ∩ C) (B ∩ C) (b) If A ⊆ B then A ∩ (X (B ∩ C)) = ∅ 2. Let X be a set and A be a subset of X. The indicator function of A, denoted χA, is the function from X to {0, 1}, given by χA(x) = 1 if x ∈ A 0 if x ∈ X A (Symbol χ is a Greek letter chi. )(a) Give an explicit example of sets X and A where the function χA : X → {0, 1} is surjective. (b) Give an explicit example of sets X and A where the function χA : X → {0, 1} is injective. (c) Prove that for all subsets A and B of X, and all x ∈ X, χA∪B(x) = χA(x) + χB(x) ) χA∩B(x) (d) Give two explicit examples, one for each of the following statements to hold: 1. χA∪B = χA + χB 2. χA∪B = χA + χB .For parts (a), (b) and (d), you should give explicit sets X, A and B, and explain why the function satisfifies given statement for this choice of sets.