If the exercise has parts (a), (b), (c), etc., label each part of your solution and include the exercise questions for these parts as well. Provide any formulas you'll use in your computation(s) or summarize your strategy in a sentence prior to any computations. This will help the reader know why you're performing the computation. Show all steps using material from this class. If you use material from another class that isn't a prerequisite, provide the relevant definition or theorem so that someone from this class could learn from your solution. Think about this step as showing me how much of the material you've learned, or write a solution so that another student in class who doesn't know how to solve the exercise could learn from your solution. Answers without justification will receive 0 points. Write a conclusion either using a sentence (preferred for an application exercise, like a word problem) or symbolically to summarize what your work shows.

2. Let a1,42,...,@, R. Answer the following questions about properties of arithmetic of real numbers. (a) (f) Let A = {1,2,3,...n}, n Z*, and a: A A be an automorphism on A (an automor- phism is a bijection of a set onto itself). How many possible functions 7 are there? Consider }"_, a;. We often use the property of commutativity a + b = b + a without even thinking about it when computing sums. Since addition is a binary operation, we can only add two numbers at a time. Use induction to prove that ", ai = 7), On(i)- State and prove a claim similar to the claim in part 2b about multiplication of real numbers. How many ways possible products can be computed using two distinct elements from the set {a, a2, ...,a@,,} and commutativity? In other words, we will count aja; and aja;, i j, as distinct products here. Note also that even if aja; = a,a, for some i,j,k,1 A, i Fj, k #l, we will count aja; and a,a; as two distinct products. How many ways possible products can be computed using three distinct elements from the set {a1, a2, ...,@n} and commutativity? How many ways possible products can be computed using 2 to n distinct elements from the set {a1, a2, ...,@n} and commutativity