Math 216 Exam 2 (150 points) Name _____________________

Homework points (max of 50) ________

1. Questions on sequences, sums, and products. (3 points each)

a. Write the first three terms of the following sequence:

b. Give an explicit formula for the sequence whose initial terms are as follows:

c. Compute the following summation:

d. Compute the following product:

e. Write the following using summation notation:

2. True or false. Circle T if the statement is always true, otherwise circle F. (13 points)

T F 3 Î {1, 2, 3, 4, 5}

T F 3 Í {1, 2, 3, 4, 5}

T F {2, 4} Í {1, 2, 3, 4, 5}

T F {a, t, e} Í {e, a, t}

T F {1} Î {{0, 1}, {2}, {3}}

T F {a, t, e} = {e, a, t} Ç {c, a, n, d, y}

T F The power set of a set with n elements contains n! elements.

3. Find A ´ B where A = {a, b} and B = {c, d}. (3 points)

4. Disprove the following claim by providing a counterexample. (3 points)

For all sets A, B, and C, if A is not a subset of B and B is not a subset of C, then A is not a subset of C.

5. Fill in the blank. (5 points)

Claim: For all sets A and B, A – B Í B^C.

Proof: Suppose A and B are any sets and x Î _____________. [We must show that __________________.] By the definition of set difference, _______________ and ___________________. In particular, x Ï B, Therefore, by definition of complement, __________________ [as was to be shown].

6. Using the handout provided, supply the appropriate reason for each step in the derivation. (7 points)

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

7. A person buying a personal computer system is offered a choice of three models of the basic unit, two models of keyboard, and four models of printer. How many distinct systems are there to choose from? (4 points)

8. Suppose that in a certain state, all automobile license plates have four letters followed by three digits.

a. How many different license plates are possible? (3 points)

b. What is the probability that a license plate would begin with A and end with 0? (3 points)

c. How many license plates could begin with LYCO? (3 points)

d. How many license plates are possible if all letters and digits are distinct? (3 points)

e. How many license plates are possible that have at least one repetition of a letter or digit? (3 points)

9. In a different state, license plates consist of from one to three letters followed by four digits. (5 points)

a. How many different license plates can the state produce? (3 points)

b. Suppose 85 letter combinations are not allowed because of their potential for giving offense. How many different license plates can the state produce? (3 points)

10. Classify each of the following appropriately? (1 point each)

Is it a function? yes no

Is it one-to-one? yes no

Is it onto? yes no

Is it a function? yes no

Is it one-to-one? yes no

Is it onto? yes no

11. How many integers from 0 through 60 must you pick in order to be sure that at least one is even? (2 points)

12. How many cards must you pull from a standard deck to ensure that you have at least 3 of the same suit? (2 points)

13. Let f: R ® R with for all real numbers x ¹ 0. Prove that f(x) is one-to-one. (5 points)

14. Let f: R ® R with for all real numbers x ¹ 0. Prove that f(x) is not onto by providing a counterexample. (5 points)

15. Let f: R ® R with for all real numbers. Find the inverse function f ^-1. (5 points)

16. Use mathematical induction to prove the following claim: (10 points)

Claim: 3²ⁿ – 1 is divisible by 8 for all integers n ³ 0.

Proof (by mathematical induction):