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

_{}

d. ** Compute** the following

_{}

e. Write the following using ** summation notation**:

_{}

2. ** True or false**.
Circle T if the statement is

T F 3
ฮ {1, 2, 3, 4, 5}

T F 3
อ {1, 2, 3, 4, 5}

T F {2,
4} อ {1, 2, 3, 4, 5}

T F {a,
t, e} อ {e, a, t}

T F {1}
ฮ {{0, 1}, {2}, {3}}

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

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

3. __Find A ____ด__** B** where A = {a, b} and B = {c, d}. (3 points)

4. ** Disprove **the following claim by providing a

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

5. Fill in the blank. (5 points)

** Claim**: For all sets

** Proof**: Suppose

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

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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* ฎ *R* with _{} for all real numbers x
น 0. ** Prove** that

14. Let *f: R* ฎ *R* with _{} for all real numbers x
น 0. ** Prove** that

15. Let *f: R* ฎ *R* with _{} for all real
numbers. ** Find the inverse function f ^{-1}**. (5 points)

16. Use ** mathematical
induction** to

__Claim__: 3^{2n} 1 is divisible by 8 for all
integers *n* ณ 0.

__Proof__ (by mathematical induction):