Review For Final Examination (Spring 2005)
Mathematics 225 (Discrete Mathematics)
1. Write down the truth tables for: and, or, exclusive or, implies, if and only if, and not.
2. The Universe class below represents American women over 21:
A = those who have developed breast cancer
B = those who work full time
C = those who are college educated
(Note: The complement of X is indicated by X '.)
(Use a Venn diagram with three "circles" and eight rights, labeled from 1 to 8.)
a. Which region(s) represents
b. Which regions represent:
c. Give a verbal description of region 7.
d. Give a verbal description of regions 4 and 6 as a group.
e. Give a symbolic description of regions 1,4, 5, and 6 collectively.
f. Give a symbolic description of region 5.
g. Which region (s) are represented by (A union B) intersect C '.
3. Given: x1 = 2, x2 = -4, x3 = 5, x4 = -2, and x5 = 0
and y1 = 3, y2 = -4, y3 = 6, y4 = 7, and y5 = 2.
a. Compute the mean for the x values.
b. Compute the mean for the y values.
c. Compute:
d.
e.
f.
4. Reindex the sum below so that it starts at j = 3:
5. Suppose that U = { a, e, i, o, u, y }
and A = { a, e, y }, B = { i, e, o } and C = { i, u, y }
a. A B
b. A ' C ' B
c. B C '
d. (B C ' ) ' A
6.
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?
7. Compute:
a. 3!
b. 6! - 3!
c. 6P3
d. 17C3
e. 76C73
8.
A Congressional committee has 5 women Republicans, 2 women Democrats, 5 male Republicans and 6 male Democrats.
a. How many ways are there to chose a chairperson and a vice chairperson?
b. How many ways are there to choose a subcommittee of 5 members?
c. How many ways are there to choose a subcommittee of 5 members how are not women?
d. How many ways are there to choose a subcommittee of 5 members who are all Democrats?
e. How many ways are there to choose a subcommittee of 5 members who are all women or all democrats?
9. Passwords are formed from decimal digits and letters, where the letters can be upper or lower case:
a. How many passwords of length 3 are possible if repeats are allowed?
b. How many passwords of length 3 are possible if repeats are not allowed?
c. How many passwords of length 4 are possible if the digit 0 can not be used to start or end the password?
d. How many passwords of length 4 are possible if all the symbols used are identical?
10. Find the least common multiple and greatest common division of:
a. 36, 120
b. 420, 98
by
i. Using prime factorization.
ii. Euclidean algorithm.
11. Find the quotient and remainder when:
a. 134 is divided by 65
b. -135 is divided by 10
c. -46 is divided by 3
11. Find the inorder, postorder, and preorder traversal of the tree below:
12. a. Use Kruskal's algorithm to find a minimum cost spanning tree.
b. Use Prim's algorithm to find a minimum cost spanning tree starting at vertex f. Repeat starting at vertex d.
13. Ignoring the weights on the edges in the graph in 12., construct a depth first search tree and a breadth first search tree starting at vertex e. Repeat for vertex b.
14. Find the distances (disregarding the edge weights) between the vertex a and every other vertex for the graph in 12.
15. Find the minimal, maximal, least, and greatest elements in Hasse diagram below if such elements exist.