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Question 1. Let A, B C R. Prove that if there exists a strictly monotone function f : A - B, then there exists a subset C C B with same cardinality as A. Question 2. For each of the following sets, decide whether it is finite, countably infinite, or uncount- ably infinite. Explain your answer briefly. (1) P(N) (2) {1, 2, 3, 4,". . }n [0.03, 1] (3) (0,00) (4) P(Z) (5) (2,3) (6) Nn (-oo, 1000) (7) RAP(R) Question 3. Fill the blanks (-) with E or C. You do not need to justify. Let A = {1, 2, {1, 2} }. (1) 0 Z (2) {{1}} P(A) (3) {1, {1, 2} } P(A) (4) {1, 2} P(A) (5) {{1,2} } P(A) (6) N P(Z) (7) P(N) P(Z) Question 4. Prove that N x Q is a countable set. Question 5. Let S be the set of all finite sequences of letters a, b, c. For example, the following are elements of S : abaaccbcbcaaa, ababe, cccaa. Is S countable? Justify your