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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.