**Review for Examination II (Fall, 2006)
**

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 x

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

a. a

b. a

c. a

5. Solve the recursions below assuming that a

a. a

b. a

6. Solve the recursion below assuming a

a

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! - 4!

b. 5!

c.

d.

e.

9. An ice cream parlor offers sundaes with 6 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 1?

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 three ones and are even?

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, 8 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 5 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 and John can not 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 two 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 ) = 2

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. wheels

b. steal

c. steel

d. starter

18.

19. Given f(x) = -x^{3}; g(x) = -4x + 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)

20. Give some examples of relations involving real numbers.