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: x_{1} = 2, x_{2} = -4, x_{3} = 5, x_{4} = -2, and x_{5} = 0

and y_{1} = 3, y_{2} = -4, y_{3} = 6, y_{4} = 7, and y_{5} = 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. _{6}P_{3}

d. _{17}C_{3}

e. _{76}C_{73}

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.