Review Examination I

Discrete Mathematics

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

(Numbers which indicate no base are to be interpreted as base 10.

1. Perform the indicated computations:

a, (121)_{10} = (?)_{2}

b. (221)_{10} = (?)_{2}

_{
}c. (122)_{10} = (?)_{2
}d. (11100111)_{2} = (?)_{10}

e. (11011100111)_{2} = (?)_{10}

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

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

4. Perform the indicated computations:

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

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

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

d. (145)_{6} = (?)_{2}

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

a. List the elements of M.

b. Is { 2 } 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, y }

C = { 1, y, z }

Compute:

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

a.

b.

c.

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

(Note: U is the universe set.)

a.

b.

c.

d.

6.

U is the set of American college students and X is the set of student who are HIV positive, Y is the set of students who were born in the United States and Z is the set of students who were are graduates of American High Schools.

a. Give a verbal description of region 5

b. Give a verbal description of regions 6, 5.

c. Give a verbal description of regions 7, 4, 5, and 1.

d. Give an algebraic description of regions 3.

e. What region or regions is given by ?

f. Give a verbal description of ?

g. What region(s) corresponds to American college students who are HIV positive?

h. What region (s) corresponds to American college students who are not born in the United States?

i. What regions (s) correspond to American college students who are not HIV positive, were born in the United States and did not graduate from American high schools?

7. Determine a_{4} when a_{1} = -3 and a_{n+1} = a_{n} + n^{2}.

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

9. Find the prime factorizations of a. 112, b. 210 c. 350

10. Find the gcd of a. 34 and 20; b. 50 and 70; c. 24 and 36.

11. Find the lcm of: a. 24 and 40; b. 30 and 24; c. 20 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 19

c. 23 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 20

b. 122 is divided by 50

c. -45 is divided by 7

d. -41 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. 17 ? mod 5

c. 122 ? mod 80

d. -19 ? 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