PLEASE TYPE ONLY***

THEORY FOUNDATION SUBJECT***

Exercise 3.3.4: Properties of functions on strings and power sets.

For each of the functions below, indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why.

(e)

Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8}. f: P(A) {0, 1, 2, 3, 4, 5, 6, 7, 8}. For X A, f(X) = |X|. Recall that for a finite set A, P(A) denotes the power set of A which is the set of all subsets of A.

(f)

Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8}. f: P(A) P(A). For X A, f(X) = A-X. Recall that for a finite set A, P(A) denotes the power set of A which is the set of all subsets of A.

(g)

Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8} and let B = {1}. f: P(A) P(A). For X A, f(X) = X - B. Recall that for a finite set A, P(A) denotes the power set of A which is the set of all subsets of A.

(h)

A = {a, b, c}, h: P(A) P(A). For X A, h(X) = X {a}.

(i)

A = {a, b, c}, h: P(A) P(A). For X A, h(X) = X {a}.