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Discrete Mathematics: Long Examination 2

Total questions: 50

Worksheet time: 4hrs 30mins

Name
Class
Date
1.

What is the set builder notation equivalent of {-4, -3, -2, -1, 0, 1, 2, 3, 4}

a)

{x | x is an integer such that x = 0}

b)

{x ∈ ℝ | x > -4}

c)

{x | x is real number less than or equal to 4}

d)

{x ∈ Z | -5 < x < 5}

2.

Which of the following is true according to the statement:

A = the set of airline flights from Manila to Cebu. B = the set of nonstop flights from Manila to Cebu

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊆ B

3.

Which of the following is true according to the statement: 

A = the set of people who speak Filipino.  B =  the set of people who can speak Spanish

a)

A ⊆ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊄ B

4.

Which of the following is true according to the statement:  

A = the set of birds. B = the set of living creatures that can fly

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊆ B

5.

Which of the following is true according to the statement: 

 A = the set of people who can speak Tagalog. B = the set of people who can speak Tagalog with a Batangas accent.

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

A ⊆ B

d)

B ⊆ A

6.

Which of the following is true according to the statement:  

A = the set of fruits. B = the set of citrus fruit

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

A ⊆ B

d)

B ⊆ A

7.

Which of the following is true according to the statement: 

A = the set of students studying discrete mathematics. B = the set of students studying Programming

a)

A ⊆ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊄ B

8.

Which of the following is true according to the statement: 

A = the set of students of Pamantasan ng Lungsod ng Maynila. B = the set students at PLM CET IT

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

A ⊆ B

d)

B ⊆ A

9.

Which of the following is true according to the statement:

A = the set of students at PLM living in Mandaluyong. B = the set of students at PLM living in Bulacan

a)

A ⊆ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊄ B

10.

Which of the following is true according to the statement: 

A = the set of Filipino basketball players playing in Japan League. B = the set of Filipinos

a)

A ⊄ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊆ B

11.

Which of the following is true according to the statement:  

A = the set of teachers at PLM. B = the set of teachers at PLM CET

a)

A ⊆ B

b)

A ⊆ B and B ⊆ A

c)

B ⊆ A

d)

A ⊄ B

12.

x ∈ {x}

a)

true

b)

false

13.

{x} ⊆ {x}

a)

true

b)

false

14.

{x} ∈ {x}

a)

true

b)

false

15.

{x} ∈ {{x}}

a)

true

b)

false

16.

∅ ⊆ {x}

a)

true

b)

false

17.

∅ ∈ {x}

a)

true

b)

false

18.

∅ ∈ {∅}

a)

true

b)

false

19.

∅ ∈ {∅,{∅}}

a)

true

b)

false

20.

{∅} ∈ {∅}

a)

true

b)

false

21.

{∅} ∈ {{∅}}

a)

true

b)

false

22.

{∅} ⊂ {∅,{∅}}

a)

true

b)

false

23.

{{∅}} ⊂ {∅, {∅}}

a)

true

b)

false

24.

 {{∅}} ⊂ {{∅},{∅}}

a)

true

b)

false

25.

0 ∈ ∅

a)

true

b)

false

26.

 ∅ ∈ {0}

a)

true

b)

false

27.

There are ____ elements in A x B if A has m elements and B has n elements

4 lines
28.

There are ____ elements in A x B x C if A has m elements, B has n elements and C has p elements 

4 lines
29.

There are ____ elements in An when A has m elements and n is a positive integer.

4 lines
30.

The truth set of R(x):2x+1=0 is _____.

4 lines
31.

Let A = {1,2,3,4,5} and B = {0,3,6}. The A ∪ B is _________.

4 lines
32.

Let A = {1,2,3,4,5} and B = {0,3,6}. The A−B is ________.

4 lines
33.

Let A = {1,2,3,4,5} and B = {0,3,6}. The A∩B is _______. 

4 lines
34.

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g}. The A∪B is ______.

4 lines
35.

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g}. The A∩B is ______.

4 lines
36.

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g}. The B−A is ______.

4 lines
37.

Find the domain and range of the following function. Note that in each case, to find the domain, determine the set of elements assigned values by the function:

The function that assigns to each nonnegative integer its last digits

4 lines
38.

Find the domain and range of the following function. Note that in each case, to find the domain, determine the set of elements assigned values by the function:

 

The function that assigns the next largest integer to a positive integer

4 lines
39.

Find the domain and range of the following function. Note that in each case, to find the domain, determine the set of elements assigned values by the function:

 

The function that assigns to a bit string the number of one bits in the string

4 lines
40.

Find the domain and range of the following function. Note that in each case, to find the domain, determine the set of elements assigned values by the function:

 

The function that assigns to a bit string the number of bits in the string

4 lines
41.

Find the domain and range of the following function. Note that in each case, to find the domain, determine the set of elements assigned values by the function:

 

The function that assigns to each bit string the number of ones in the string minus the number of zeros in the string

4 lines
42.

Let O = {a,c,j,k} and N = {e,z}. Find O x N

4 lines
43.

If f and f ◦ g are one-to-one, does it follow that g is one-to-one? Justify your answer.

4 lines
44.

Let O = {a,c,j,k} and N = {e,z}. Find N x O

4 lines
45.

Find the truth set of each of the predicates where the domain is the set of positive integers. (You may write the set in standard form or set builder notation):

P(x): x3 ≥ 1

4 lines
46.

Find the truth set of each of the predicates where the domain is the set of positive integers. (You may write the set in standard form or set builder notation):

R(x): x < x2

4 lines
47.

Find the truth set of each of the predicates where the domain is the set of positive integers. (You may write the set in standard form or set builder notation):

Q(x): x2 = 2

4 lines
48.

Suppose that the universal set is U = {1,2,3,4,5,6,7,8,9,10}. Express the following set sets with bit strings where the i-th bit in the string is 1 if i is in the set and 0 otherwise:

{3,4,5}

4 lines
49.

Suppose that the universal set is U = {1,2,3,4,5,6,7,8,9,10}. Express the following set sets with bit strings where the i-th bit in the string is 1 if i is in the set and 0 otherwise:

{1,3,6,10}

4 lines
50.

Suppose that the universal set is U = {1,2,3,4,5,6,7,8,9,10}. Express the following set sets with bit strings where the i-th bit in the string is 1 if i is in the set and 0 otherwise:

{2,3,4,5,8,9}

4 lines