Question: 2. Let X be the set {a, b, c,..., z). Give answers to each of the following questions, justifying your answer in each case.

2. Let X be the set {a, b, c,..., z). Give answers to each of the following questions, justifying your answer

2. Let X be the set {a, b, c,..., z). Give answers to each of the following questions, justifying your answer in each case. (You don't need to simplify arithmetic expressions.) (a) How many functions are there which map from X to X? (b) How many distinct total orders can be defined on X? (c) For each function f in the set of functions from X to X, consider the relation that is the symmetric closure of the function f. Let us call the set of these symmetric closures Y. List at least two elements of Y. (d) Suppose R is some partial order on X. What is the smallest possible cardinality R could have? What is the largest?

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