This problem set is all proofs. Most of them are one line long if you are carefu

Question

This problem set is all proofs. Most of them are one line long if you are careful about referring to previously established results. You should consider this problem set as a diagnostic exam for this course. If you have serious difficulties then your proof technique is not sufficient for the course. By now you should have read Notes on Functions, Finite Sets and Counting, which we will refer to by NFC. We repeat some of the material. Let X and Y. The set of all functions mapping X to Y, which by definition is a subset of P(X times Y), is denoted by F(X, Y). An alternate notation is Y^X The next sequence of problems is a slightly different development of the results of BB Section 2.1 or C ch 6. Thus you should not refer to results in either BB or C as part of your proofs. (1) Problem Take X and Y sets. (a) Let X = phi what set is X times Y. (b) Let Y = phi what set is X times Y. (c) X = phi what set is F(X, Y) (d) Let X notequalto phi, Y = phi what set is F(X, Y) Let A and B be sets. Take a function mu: A rightarrow B and functions lambda: B rightarrow A, rho: B rightarrow A Left Inverse: The map lambda: B rightarrow A is a left inverse for mu if lambda compositefunction mu = Id_A. Right Inverse: The map rho: B rightarrow A is a right inverse for mu if mu compositefunction rho = Id_B Inverse: (1) If lambda is a left inverse for mu then mu is a right inverse for lambda. (2) If rho is a right inverse for mu then phi is a left inverse for rho.

Explanation / Answer

X x Y is an empty set because X is an empty set .An empty set can only point to another empty set

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