Review for Examination II (Spring, 2007)

Mathematics 225 (Discrete Mathematics)

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, New York 11451

email: malkevitch@york.cuny.edu

1. Compute:

a.

b.

c.

2. Suppose x1 = -2, x2 = -1, x3 = 1, and x4 = 5 and y1 = 2, y2 = -2, y3 = 0, and y4 = 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 a3, and a4 assuming that a0 = -1.

a. an+1 = 5an + 2n + 5.

b. an+1 = -2an + n - 3

c. an+1 = -3an + 2(4n).

5. Solve the recursions below assuming that a0 = -4:

a. an+1 = -4an

b. an = 3an-1

6. Solve the recursion below assuming a0 = +3, a1 = -1

an+2 = -4an+1 - 3an

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! -5!

b. 5!

c. 6P3

d. 17C3

e. 76C73

9. An ice cream parlor offers sundaes with 6 types of syrup, 4 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 10?

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 and end with a 0?

e. How many binary sequences of length 7 are there that begin with two ones or end with 00? (Be careful.)

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 11 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 5 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. Suppose g(x) = 2x - 3, and h (x ) = 2x

a. Find g (h(2)) and h(g(2).

b. Fine h(g(3)) and g(h(3))

c. Find h(g(-1)) and g(h(-1)).

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.

19. Given f(x) = -x3; g(x) = -3x + 12

i. Evaluate:

a. f(0), f(-2), f(1/2), g(0), g(-2), g(1/2)

b. f(3x); g(x + h)

c. fog(2), gof(-2)

ii. Find: The inverse of f(x)

iii. Find: The inverse of g(x)