Let X be a set and A be a subset of X. The indicator function of A, denoted ?A, is the function from X to {0, 1}, given by

? 1 if x?A ?A(x)= 0 if x?X\A

(Symbol ? is a Greek letter chi.)

(a)Give an explicit example of sets X and A where the function ?A : X ? {0, 1} is

surjective.

(b)Give an explicit example of sets X and A where the function ?A : X ? {0, 1} is injective.

(c)Prove that for all subsets A and B of X, and all x?X, ?A?B(x) = ?A(x) + ?B(x) ? ?A?B(x)

(d)Give two explicit examples, one for each of the following statements to hold: 1. ?A?B = ?A + ?B

2. ?A?B?=?A+?B.

For parts (a), (b) and (d), you should give explicit sets X, A and B, and explain why

the function satisfies given statement for this choice of sets.

2. Let X be a set and A be a subset of X. The indicator function of A, denoted XA, is the function from X to {0, 1}, given by 1 if TEA XA(20) = 0 if TEX A (Symbol x is a Greek letter chi.) (a) Give an explicit example of sets X and A where the function XA : X -+ {0, 1} is surjective. (b) Give an explicit example of sets X and A where the function XA : X - {0, 1} is injective. (c) Prove that for all subsets A and B of X, and all r E X, XAUB() = XA(X) + XB(X) - XAnB(X) (d) Give two explicit examples, one for each of the following statements to hold: 1. XAUB = XA + XB 2. XAUB # XA + XB. For parts (a), (b) and (d), you should give explicit sets X, A and B, and explain why the function satisfies given statement for this choice of sets