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 x1 = -2, x2 = 4, x3 = 0, and x4 = 5 and y1 = 3, 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 = -3:
a. an+1 = -3an
b. an = 4an-1
6. Solve the recursion below assuming a0 = +2
an+1 = -5an + 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. 6P3
d. 17C3
e. 76C73
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 n3 + 5n is a multiple of 6
Hint: If 6 divides k3 + 5k then we can write k3 + 5k = 6w where w is some integer.
15. a. Write down the relation associated with the digraph below:
