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set theory

Total questions: 53

Worksheet time: 3hrs 45mins

Name
Class
Date
1.

Which of the following sentences are propositions? For those that are, indicate the truth value.

a)

a. 123 is a prime number.

b)

b. 0 is an even number.

c)

c. 푥

d)

d. Multiply 5푥+2 by 3.

e)

e. What an impossible question!

2.

State the negation of each of the following statements.

4 lines
3.

Let 푝: 15 is an odd number. 푞: 21 is a prime number.

4 lines
4.

Complete the following truth table.

4 lines
5.

For statements 푝, and 푟, use a truth table to show that each of the following pairs of statements is logically equivalent.

a)

a. (푝∧푞)⟺푝 and 푝⟹푞.

b)

b. 푝⟹(푞∨푟) and 푞⟹(푝∨푟).

c)

c. (푝∨푞)⟹푟 and (푝⟹푞)∧(푞⟹푟).

d)

d. 푝⟹(푞∨푟) and (푟)⟹(푝⟹푞).

6.

For statements 푝, q, and 푟, show that the following compound statements are tautology.

a)

a. 푝⟹(푝∨푞).

b)

b. (푝∧(푝⟹푞))⟹푞.

c)

c. ((푝⟹푞)∧(푞⟹푟))⟹(푝⟹푟).

7.

Write the contrapositive and the converse of the following conditional statements.

a)

a. If it is cold, then the lake is frozen.

b)

b. If Solomon is healthy, then he is happy.

c)

c. If it rains, Tigist does not take a walk.

8.

Show that (푝∧푞)∧(푝∧푞) is a contradiction.

4 lines
9.

Find the truth value of each of the following statements: (푝∨푞)∨푟, 푝∨(푞∨푟), 푟⟹(푠∧푝), 푝⟹(푟⟹푠), 푝⟹(푟∨푠), (푝∨푟)⟺(푟∧Ø푠), (푠⟺푝)⟹(Ø푝∨푠), (푞∧Ø푠)⟹(푝⟺푠), (푟∧푠)⟹(푝⟹(Ø푞∨푠)), (푝∨Ø푞)∨푟⟹(푠∧Ø푠).

4 lines
10.

What can be said about the value of 푝∧푞⟺푝∨푞?

4 lines
11.

What can be said about the values of 푝⟺푞 and 푝⟺푞?

4 lines
12.

Construct the truth table for each of the following statements: 푝⟹(푝⟹푞), (푝∨푞)⟺(푞∨푝), 푝⟹(푞∧푟), (푝⟹푞)⟺(푝∨푞), (p⟹(q∧r))∨(p∧q), (푝∧푞)⟹((푞∧푞)⟹(푟∧푞)).

4 lines
13.

Determine whether the information given is sufficient to decide the truth value of the statement: (푝⟹푞)⟹푟, where 푟=푇.

4 lines
14.

Determine whether the information given is sufficient to decide the truth value of the statement: 푝∧(푞⟹푟), where 푞⟹푟=푇.

4 lines
15.

Determine whether the information given is sufficient to decide the truth value of the statement: 푝∨(푞⟹푟), where 푞⟹푟=푇.

4 lines
16.

Determine whether the information given is sufficient to decide the truth value of the statement: (푝∨푞)⟺(푝∧푞), where 푝∨푞=푇.

4 lines
17.

What is the truth value of ¬(p∨q)⟺(¬p∧¬q)?

4 lines
18.

What is the truth value of (p⟹q)⟹(¬q⟹¬p) when q=T?

4 lines
19.

Determine the truth value of (p∧q)⟹(p∨s) when p=T and s=F.

4 lines
20.

For the open statement (x,):x/y<1 with domains A={2,3,5} and B={2,4,6}, state the quantified statement (∀x ∈ A)(∃y ∈ B)P(x, y) in words.

4 lines
21.

Use the truth table method to show that the following argument forms are valid. i. 푝⟹푞, 푞 ├ 푝.

4 lines
22.

Use the truth table method to show that the following argument forms are valid. ii. 푝⟹푝,,⟹푞 ├ 푟.

4 lines
23.

Use the truth table method to show that the following argument forms are valid. iii. 푝⟹푞, 푟⟹푞 ├ 푟⟹푝.

4 lines
24.

Use the truth table method to show that the following argument forms are valid. iv. 푟∨푠,( 푠⟹푝)⟹푟 ├ 푝.

4 lines
25.

Use the truth table method to show that the following argument forms are valid. v. 푝⟹푞, 푝⟹푟,⟹푠├ 푞⟹푠.

4 lines
26.

Give formal proof to show that the following argument forms are valid. a. 푝⟹푞, 푞 ├ 푝.

4 lines
27.

Prove the following are valid arguments by giving formal proof. If the rain does not come, the crops are ruined and the people will starve. The crops are not ruined or the people will not starve. Therefore, the rain comes.

4 lines
28.

If the team is late, then it cannot play the game. If the referee is here then the team can play the game. The team is late. Therefore, the referee is not here.

4 lines
29.

Which of the following are sets?

4 lines
30.

Which of the following sets can be described in complete listing, partial listing and/or set-builder methods? Describe each set by at least one of the three methods.

4 lines
31.

Write each of the following sets by listing its elements within braces.

4 lines
32.

Find the truth value of each of the following. Is 15 an element of set A?

4 lines
33.

Find the truth value of each of the following. Is the empty set a subset of any set A?

4 lines
34.

For each of the following set, find its power set. Set: {a, b}

4 lines
35.

How many subsets and proper subsets do the sets that contain exactly 1,2,3,4,8,10 and 20 elements have?

4 lines
36.

If n is a whole number, use your observation in Problems 6 and 7 to discover a formula for the number of subsets of a set with n elements. How many of these are proper subsets of the set?

4 lines
37.

Is there a set A with exactly the following indicated property? Only one subset

4 lines
38.

Find the truth value of each of the following. Is the empty set an element of the power set of A?

4 lines
39.

For any three sets A, B and C, prove that if A is a subset of B and B is a subset of C, then A is a subset of C.

4 lines
40.

If B is a subset of A, A intersection B' = {1,4,5} and A union B = {1,2,3,4,5,6}, find B.

4 lines
41.

Which of the following implies that 푝∨푞 is false?

a)

푝∨푞

b)

푝∨푞

c)

푝∧푞

d)

푝⟹푞

42.

What is the transformation of: f(x) = 1x+3f\left(x\right)\ =\ \frac{1}{x}+3  

a)

Left 3

b)

Right 3

c)

Up 3

d)

Down 3

43.

What is the transformation of: f(x) = 1x+3f\left(x\right)\ =\ \frac{1}{x}+3  

a)

Left 3

b)

Right 3

c)

Up 3

d)

Down 3

44.
Simplify.
a)
1/(n-7)
b)
n-7
c)
(n-6)/(n-7)
d)
n-6
45.
Simplify.
a)
2p
b)
p/2
c)
2/p
d)
4p
46.
Simplify.
a)
2p
b)
p/2
c)
2/p
d)
4p
47.
What are the asymptotes?
a)
x=-1 y = -3
b)
x=1 y = -3
c)
x=-1 y =3
d)
x=1, y=3
48.
From the above Venn diagram, what is the set S ∩ T?
a)
{casey, drew, jade, glen}
b)
{alex, casey, drew,hunter}
c)
{casey, drew}
d)
{drew, jade}
49.

If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∪ B?

a)

A ∪ B = {3, 5, 7}

b)

A ∪ B = {2, 3, 5, 7}

c)

A ∪ B = {2, 3, 5, 7, 9}

d)

A ∪ B = {1, 2, 3, 5, 7, 9}

50.

What does this symbol ( ∪ ) represent in set theory?

a)

Union

b)

Intersection

c)

Disjointed

d)

Subset

51.
The Universal Set = { -4, 3, -2, -1, 0, 1, 2, 3 ,4} and A = {0}.
What is the complement of A?
a)
{-4, -3, -2, -1, 0, 1, 2, 3}
b)
{-3, -2, -1, 1, 2, 3}
c)
{-4, -3, -2, -1, 1, 2, 3, 4}
d)
{-4, -3, -2, -1, 1, 2, 3}
52.

If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∩ B?

a)

{3, 5, 7}

b)

{2, 3, 5, 7}

c)

{2, 3, 5, 7, 9}

d)

{1, 2, 3, 5, 7, 9}

53.

If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∩ B?

a)

{3, 5, 7}

b)

{2, 3, 5, 7}

c)

{2, 3, 5, 7, 9}

d)

{1, 2, 3, 5, 7, 9}

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