**Review: Examination I**

*Discrete Mathematics*

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York 11451

1. In the diagram below, X represents students who are majoring in mathematics, Y represents students who are Juniors, and Z represents students who are married.

a. Write down a verbal and symbolic expression for region 3.

b. Write down a verbal and symbolic expression for regions 6 and 8.

c. Write down a verbal and symbolic expression for regions 3, 4, 7 and 8.

d. What region(s) represent American college students who are not mathematics majors, not juniors, but are married?

e. Write a verbal description of what represents.

f. Write a verbal description of what represents.

g. What regions are represented by American college students who are married or not mathematics majors?

2. Given the set M = { 1, 2, 3, { 3 } }

a. List the elements of M.

b. Is { 1 } an element of M?

c. Is { 2, 3 } a subset of M?

d. List all the subsets of M.

Given that U = { 1, 2, 3, x, y, z }

A = { 1, 3 }

B = { 3, x, z }

C = { 1, y, z }

Compute:

(Note: Here X' is used for complement of X.)

a.

b.

c.

3. Use a Venn Diagram to determine if the following statements are true or false:

(Note: U is the universe set.)

a.

b.

c.

d.

4. Perform the indicated computations:

a, (122)_{10} = (?)_{2}

b. (211)_{10} = (?)_{2}

_{
}c. (122)_{10} = (?)_{2
}d. (11101111)_{2} = (?)_{10}

e. (11010100111)_{2} = (?)_{10}

5. Count from 39 to 50 in base 2 (i.e. write these numbers down in binary notation.

6. Explain briefly the difference between the term "number" and "digits (base b)."

7. Perform the indicated computations:

a. (123)_{4} = (?)_{10}

b. (122)_{3} = (?)_{10}

c. (135)_{6} = (?)_{10}

d. (145)_{10} = (?)_{4}

e. (A3)_{16} = (?)_{10}

8. State the two DeMorgan Laws in their set theory form.

9. Find the prime factorizations of a. 122, b. 210 c. 350

10. Find the gcd of a. 54 and 20; b. 50 and 70; c. 24 and 84.

11. Find the lcm of: a. 24 and 40; b. 30 and 24; c. 18 and 34.

12. Use the Euclidean algorithm to write the gcd of d of the numbers a and b:

a. 14 and 22

b. 24 and 17

c. 41 and 37

in the form: d = xa + by where x and y are integers.

13. Find the remainder (which must be a non-negative integer less than the divisor) when:

a. 132 is divided by 22

b. 122 is divided by 50

c. -45 is divided by 8

d. -42 is divided by 20

14. Find the value of the ? which is greater than or equal to 0, positive and smaller than the modulus:

a. 12 ? mod 10

b. 18 ? mod 5

c. 132 ? mod 80

d. -18 ? mod 10

e. -16 ? mod 11

f. -55 ? mod 100

15. Are the following true or false:

a. 23 -1 mod 12

b. -13 6 mod 5

c. -13 7 mod 5

d. -13 -7 mod 6

e. -121 29 mod 23

f. 24 + 12 -2 + 11 mod 3

g. (-8)^{2} 1 mod 2

h. -(-8)^{2} 1 mod 2

i. (-2)^{3} 5 mod 7

16. Solve the congruences below for x by finding a solution where x lies between 0 and the modulus - 1.

a. 4x 1 mod 3

b. 11x 1 mod 17

c. 3x 4 mod 7

d. -2x 5 mod 11