Math 187, Spring 2018: Test III

Transcription

1 Math 187, Spring 2018: Test III Dr. Holmes April 13, 2018 The exam starts at 9 am and ends at 9:55. You may use your writing instrument and your non-graphing calculator. Cell phones must be turned off and out of sight. You may drop one of the eight numbered questions. If you do all of them, your best work will count. 1

2 1. This question has four parts. Each part involves a choice of k items taken from one of n alternatives. In some, the order in which choices are made matters; in some it does not ( order matters, or not). In some, the same alternative may be chosen more than once; in some this is not allowed ( repetitions allowed, or not). In each part, say whether order matters or not, say whether repetitions are allowed or not, set up the calculation needed to solve the problem (straightforward in each case: these are not tricky questions) and state the answer. Most of the credit is for the answers. (a) A bicycle license plate in a certain town consists of three letters. How many plates are possible? (b) A lady dressing for the evening wants to wear three of her ten rings. In how many ways can she choose which ones to wear? (c) A young woman wants to order a dozen roses at the florist: there are white roses, pink roses, and red roses. How many possible choices of how many roses to have of each color can she make? (d) A bag of Scrabble tiles contains one tile with each letter. In how many ways can one choose five tiles from the bag and arrange them into a word (it doesn t have to be a correct word) on the board? 2

3 2. Prove that the sum n i=1 2i, the sum n of the first n even numbers, is n 2 n, using mathematical induction. Label the basis step,. the induction hypothesis, and the induction goal, and clearly indicate where in your proof the induction hypothesis is used. 3

4 3. Prove by mathematical induction that 4 n 1 is divisible by three for each natural number n. Label the basis step,. the induction hypothesis, and the induction goal, and clearly indicate where in your proof the induction hypothesis is used. 4

5 4. In this problem, you may choose to do just one of the two parts. If you do both, your best work will count. If you do well on both, you might earn additional credit. (a) The sequence a n is defined as follows: a 0 = 3; a 1 = 8; a n+2 = 6a n+1 8a n. Compute the next three terms of this sequence. Derive a formula for the nth term of this sequence and check it for the terms you have computed. 5

6 (b) The sequence with the following first few terms b 0 = 0 b 1 = 0 b 2 = 3 b 3 = 12 b 4 = 30 is determined by a third degree polynomial. Find the polynomial using the method of differences. 6

7 5. For each of the following relations, carry out the following steps. If it is not a function, say why and stop (giving specific numerical examples). If it is a function, say what its domain and image are. If it is not one-to-one, say why and stop. If it is one-to-one, describe its inverse function (it will be possible to compute a formula for the inverse). (a) x 2 + y 2 = 1, x, y real numbers. (b) y = x 2 + 1, x, y real numbers. (c) y = x 3 + 1, x, y real numbers. 7

8 6. Let A = {1, 2, 3, 4, 5} and let B = {a, b}. How many functions are there from B to A? How many one-to-one functions are there from B to A? If there are any, show one as a list of ordered pairs. How many functions are there from B onto A? If there is one, show it as a list of ordered pairs. How many functions are there from A to B How many one-to-one functions are there from A to B? If there are any, show one as a list of ordered pairs. How many functions are there from A onto B? Hint: how many functions from A to B are there which are not onto B? If there is one, show one as a list of ordered pairs. 8

9 7. Two functions f and g are given as lists of ordered pairs. Compute f g and g f as sets of ordered pairs, or if one or both of them do not exist, explain. f = {(a, 1), (b, 3), (c, 1)} g = {(1, a), (2, c), (3, b)} I suggest drawing arrow diagrems to do your calculations, but don t forget to write your final answers as sets of ordered pairs in list notation, and don t forget to tell me which is which! 9

10 8. Explain why it has to be the case that if you choose eleven distinct natural numbers, two of them must have a difference divisible by ten, using the Pigeonhole Principle. 10