Review for Examination II (Spring, 2005)

prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, New York 11451

email: joeyc@cunyvm.cuny.edu

1. Compute:

a.

b.

c.

2. Suppose x_{1} = -2, x_{2} = 4, x_{3} = 0, and x_{4} = 5 and y_{1} = 3, y_{2} = -2, y_{3} = 0, and y_{4} = 2.

Compute:

a.

b.

c.

d.

e.

3. Prove using mathematical induction that for all n:

4. For each of the difference equations below determine the values of a_{3}, and a_{4} assuming that a_{0} = -1.

a. a_{n+1} = 5a_{n} + 2n + 5.

b. a_{n+1} = -2a_{n} + n - 3

c. a_{n+1} = -3a_{n} + 2(4^{n)}.

5. Solve the recursions below assuming that a_{0} = -3:

a. a_{n+1} = -3a_{n}

b. a_{n} = 4a_{n-1}

6. Solve the recursion below assuming a_{0} = +2

a_{n+1} = -5a_{n} + 4

7. Determine if the following are tautologies. If not (if possible) give one set of truth for which the logical expression is true.

a.

b.

c.

d.

8. Compute:

a. 3!

b. 5!

c. _{6}P_{3}

d. _{17}C_{3}

e. _{76}C_{73}

9. An ice cream parlor offers sundaes with 5 types of syrup, 3 toppings, and 24 flavors of ice cream. How many different kinds of sundaes can be made?

10. a. How many binary sequences of length 7 are possible?

b. How many binary sequences of length 7 are there that begin and end with 0?

c. How many binary sequences of length 7 are there have ones in odd numbered positions?

d. How many binary sequences of length 7 are there that begin with two ones?

11. Give an example of a proposition which is:

a. True

b. False

c. Truth status is not known.

12. The psychology club of a college has 10 members, 7 men.

a. How many ways are there to elect a president, a vice-president, and a treasurer for the club?

b. How many ways are there to elect a president, a vice-president, and a treasurer for the club if the president must be a girl?

c. How many ways are there to elect a president, a vice-president, and a treasurer for the club, if the vice-president and treasurer must be boys?

d. How many ways can a committee of 4 be chosen to interview the President of their college?

e. How many ways can a committee of 4 which has exactly 2 women be chosen?

f. How many ways can a committee of 4 be chosen if Mary must be on the committee?

g. How many ways can a committee of 5 be chosen in Mary and John must be on the committee?

h. How many ways are there to choose a committee of 5 if there can be at most one woman on the committee?

13. If the following result is true, prove it using mathematical induction:

14.

Prove using mathematical induction that n^{3} + 5n is a multiple of 6

Hint: If 6 divides k^{3} + 5k then we can write k^{3} + 5k = 6w where w is some integer.

15. a. Write down the relation associated with the digraph below:

b. Write down the matrix associated with the digraph above.

16. If x = { 1, 2, 3 } and Y = { 1, -1 } and Z = { 2, -3, -1 }

a. Write down the cartesian product of X and Y.

b. Write down the cartesian product of Y and X

c. Write down the cartesian product of Z and Y

d. Write down the cartesian product of X and Z.

17. How many different nonsense words can be made with the letters of the word:

a. Teeth

b. steal

c. steel

d. starter

18.