. Prove that each of the following operations is associative. (a) The intersection operation on the...

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. Prove that each of the following operations is associative. (a) The intersection operation on the power set P(X) of a set X. (Hint: You need to prove (An B) nC = An (BNC) for all A, B, C = P(X). You can do this by a sequence of "if and only if" steps: For all x E X, x € (An B) nC if and only if...if and only if x € An (BNC).) (b) The binary operation x * y = (c) The binary operation x * y = x+y=xy on Z. x + y on the interval (-1,1). 1 + xy (d) The binary operation x * y = x on a set X. . Prove that each of the following operations is associative. (a) The intersection operation on the power set P(X) of a set X. (Hint: You need to prove (An B) nC = An (BNC) for all A, B, C = P(X). You can do this by a sequence of "if and only if" steps: For all x E X, x € (An B) nC if and only if...if and only if x € An (BNC).) (b) The binary operation x * y = (c) The binary operation x * y = x+y=xy on Z. x + y on the interval (-1,1). 1 + xy (d) The binary operation x * y = x on a set X.