Consider the set S of strings defined recursively in Problem 1. Prove that for any given string in S, the number of a's in it is the same as the number of b's in it. Prove each statement using either weak, strong, or structural induction. Make sure to clearly indicate the different parts of your proof: the basis step, the inductive hypothesis, what you will show in the inductive step, and the inductive step. Make sure to clearly format your proofs and to write in complete, clear sentences. EXAMPLE: Prove that for any nonnegative integer n, Σ i = (n+1) Answer: Proof. (by weak induction) Basis step: n = 1 Σ=1 1(1+1)==1 Therefore, (n+1) when n = 1. = Inductive hypothesis: Assume that Inductive step: We will show that i=1 i=1 i= = (+1) for some integer k > 1. i= (k+1)((k+1)+1) k+1 Σ=Σ+ (κ + 1) i=1 By inductive hypothesis, k+1 Σ IME i=1 k(k+1) = +k+1 2 k(k+1)+2(k+1) = 2 (k+2)(k+1) = 2 (k+1)((k+1)+1) 2 Therefore, by weak induction, we have shown that = (n+1) for all nonnegative integers n.

(Program) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters, such as zerocount, onecount, twocount, and so forth, to 0. Then generate a large number of pseudorandom integers between 0 and 9. Each time 0 occurs, increment zerocount; when 1 occurs, increment onecount; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of time they occurred.

The implementation of a queue in an array, as given in this chapter, uses the variable count to determine whether the queue is empty or full. You can also use the variable count to return the number of elements in the queue. On the other hand, class linkedQueueType does not use such a variable to keep track of the number of elements in the queue. Redefine the class linkedQueueType by adding the variable count to keep track of the number of elements in the queue. Modify the definitions of the functions addQueue and deleteQueue as necessary. Add the function queueCount to return the number of elements in the queue. Also, write a program to test various operations of the class you defined.