Logic and Set Theory Covering Propositional Logic, Set Operations, and Venn Diagrams

Logic and Set Theory: Covering Propositional Logic, Set Operations, and

Venn Diagrams

1. If p: "It is raining" and q: "The ground is wet", which of the following represents "If it is raining,

then the ground is wet"?

a) p ∧ q

b) p ∨ q

c) p → q

d) p ↔ q

Answer: c) p → q

d) Some cats are black

Answer: b) Some cats are not black

Solution: The negation of a universal statement ("All A are B") is an existential statement ("Some A

are not B"). Therefore, "Some cats are not black" is the correct negation.

3. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

a) {1, 2, 3, 4, 5}

b) {3}

c) {1, 2, 4, 5}

d) {1, 2, 3, 3, 4, 5}

Answer: a) {1, 2, 3, 4, 5}

d) Neither true nor false

Answer: a) True

Solution: Let's evaluate step by step:

1. (T ∧ F) = F

2. (T ∨ F) = T

3. F ∨ T = T

Therefore, the final result is True.

5. If A = {x | x is an even number less than 10} and B = {x | x is a prime number less than 10}, what is

A ∩ B?

a) {2}

B = {2, 3, 5, 7}

The intersection of A and B (A ∩ B) contains elements that are in both A and B. The only such

element is 2.

6. Which of the following is logically equivalent to p → q?

a) ¬p ∨ q

b) p ∧ q

c) ¬q → ¬p

d) q → p

Answer: a) ¬p ∨ q

Solution: The conditional statement p → q is logically equivalent to its contrapositive ¬q → ¬p, and

also to the disjunction ¬p ∨ q. This is a fundamental equivalence in propositional logic.

7. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, what is A'? (A' denotes the complement of

A)

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

8. Which of the following is the converse of "If it rains, the grass gets wet"?

a) If the grass is wet, it rains

b) If it doesn't rain, the grass doesn't get wet

c) If the grass doesn't get wet, it doesn't rain

d) It rains if and only if the grass gets wet

Answer: a) If the grass is wet, it rains

Solution: The converse of a conditional statement "If p, then q" is "If q, then p". Therefore, the

converse of "If it rains, the grass gets wet" is "If the grass is wet, it rains".

9. In a Venn diagram representing A ∪ B, which region is shaded?

a) Only the region inside circle A

b) Only the region inside circle B

a) x ≥ 5 or y ≤ 3

b) x ≥ 5 and y ≤ 3

c) x < 5 or y > 3

d) x ≥ 5 and y > 3

Answer: a) x ≥ 5 or y ≤ 3

Solution: To negate a compound statement connected by "and", we negate each part and change

"and" to "or". So, the negation of "x < 5 and y > 3" is "not(x < 5) or not(y > 3)", which simplifies to "x

≥ 5 or y ≤ 3".

11. If A = {1, 2, 3} and B = {3, 4, 5}, what is A - B?

a) {1, 2}

d) {3}

Answer: a) {1, 2}

c) p → q

d) p ↔ q

Answer: b) p ∨ ¬p

Solution: A tautology is a compound proposition that is always true regardless of the truth values

of its components. p ∨ ¬p (read as "p or not p") is always true, making it a tautology.

13. If A has 5 elements and B has 7 elements, and A ∩ B has 3 elements, how many elements are in A

a) 12

c) 15

Answer: b) 9

Solution: We can use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Here, n(A ∪ B) = 5 + 7 - 3 = 9

b) If it's not sunny, then I won't go to the beach

d) If it's sunny, then I won't go to the beach

a) The union of A and B

b) The intersection of A and B

c) The Cartesian product of A and B

d) The set difference of A and B

Answer: c) The Cartesian product of A and B

Solution: A × B represents the Cartesian product of sets A and B. It is the set of all ordered pairs (a,

b) where a is an element of A and b is an element of B.

16. Which of the following is equivalent to the statement "Not all birds can fly"?

a) All birds cannot fly

b) No birds can fly

17. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is the cardinality of A ∆ B? (∆ denotes symmetric

difference)

b) 4

c) 6

Solution: The symmetric difference A ∆ B contains elements that are in either A or B, but not in

18. What is the result of the operation (A ∪ B)' ∩ (A ∩ B)'?

a) A' ∩ B'

c) (A' ∩ B) ∪ (A ∩ B')

d) ∅

Answer: a) A' ∩ B'

Solution: Using De Morgan's laws:

(A ∪ B)' = A' ∩ B'

So, (A ∪ B)' ∩ (A ∩ B)' = (A' ∩ B') ∩ (A' ∪ B')

umbrella if and only if it's raining"?

a) p → q

b) q → p

propositional logic. It means that the statement is true when both parts have the same truth value.

d) 45

Answer: a) 15

Total = Math + Physics - Both + Neither

100 = 65 + 45 - 25 + Neither

Neither = 100 - 65 - 45 + 25 = 15

21. Which of the following is a contradiction?

a) p ∧ ¬p

b) p ∨ ¬p

c) p → q

d) p ↔ q

Answer: a) p ∧ ¬p

Solution: A contradiction is a compound proposition that is always false regardless of the truth

values of its components. p ∧ ¬p (read as "p and not p") is always false, making it a contradiction.

22. If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, what is |A ∪ B|? (|X| denotes the cardinality of set X)

a) 5

c) 10

Answer: b) 8

Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

18. What is the result of the operation (A ∪ B)' ∩ (A ∩ B)'?

b) A' ∪ B'

c) (A' ∩ B) ∪ (A ∩ B')

d) ∅

Answer: a) A' ∩ B'

Solution: Using De Morgan's laws:

a) p → q

b) q → p

c) p ↔ q

d) ¬p ∨ q

Answer: c) p ↔ q

Solution: The phrase "if and only if" is represented by the biconditional operator (↔) in

propositional logic. It means that the statement is true when both parts have the same truth value.

20. In a group of 100 students, 65 study mathematics, 45 study physics, and 25 study both. How

many students study neither mathematics nor physics?

a) 15

d) 45

Answer: a) 15

Total = Math + Physics - Both + Neither

Neither = 100 - 65 - 45 + 25 = 15

Answer: b) 8

Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

The cardinality of this set is 8.

23. Which of the following is logically equivalent to ¬(p → q)?

c) ¬p ∨ q

is equivalent to p ∧ ¬q.

24. In a Venn diagram representing A ∩ B, which region is shaded?

a) Only the region inside circle A

c) The regions inside both circles A and B, including their overlap

B. In a Venn diagram, this corresponds to shading only the region where the two circles overlap.

25. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is P(A ∩ B)? (P(X) denotes the power set of X)

b) {{3}, {4}, {3, 4}}

c) {{}, {3, 4}}

d) {3, 4}

- The empty set {}

- The set itself {3, 4}

c) {1, 2, 4, 5}

or in both. Therefore, A ∪ B = {1, 2, 3, 4, 5}.

Answer: a) True

Solution: Let's evaluate step by step:

2. (T ∨ F) = T

3. F ∨ T = T

Therefore, the final result is True.

5. If A = {x | x is an even number less than 10} and B = {x | x is a prime number less than 10}, what is

A ∩ B?

a) {2}

b) {2, 3, 5, 7}

A = {2, 4, 6, 8}

The intersection of A and B (A ∩ B) contains elements that are in both A and B. The only such

element is 2.

6. Which of the following is logically equivalent to p → q?

c) ¬q → ¬p

also to the disjunction ¬p ∨ q. This is a fundamental equivalence in propositional logic.

7. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, what is A'? (A' denotes the complement of

a) {1, 3, 5, 7, 9}

Answer: a) {1, 3, 5, 7, 9}

Solution: The complement of set A (denoted A') contains all elements in the universal set U that are

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

8. Which of the following is the converse of "If it rains, the grass gets wet"?

a) If the grass is wet, it rains

Answer: a) If the grass is wet, it rains

Solution: The converse of a conditional statement "If p, then q" is "If q, then p". Therefore, the

converse of "If it rains, the grass gets wet" is "If the grass is wet, it rains".

a) Only the region inside circle A

b) Only the region inside circle B

c) The regions inside both circles A and B, including their overlap

d) Only the region where circles A and B overlap

Answer: c) The regions inside both circles A and B, including their overlap

Solution: The union of sets A and B (A ∪ B) includes all elements that are in A, or in B, or in both. In

10. What is the negation of the statement "x < 5 and y > 3"?

a) x ≥ 5 or y ≤ 3

b) x ≥ 5 and y ≤ 3

c) x < 5 or y > 3

d) x ≥ 5 and y > 3

Answer: a) {1, 2}

Solution: The set difference A - B contains all elements that are in A but not in B. In this case, 1 and

d) p ↔ q

Answer: b) p ∨ ¬p

Solution: A tautology is a compound proposition that is always true regardless of the truth values

of its components. p ∨ ¬p (read as "p or not p") is always true, making it a tautology.

13. If A has 5 elements and B has 7 elements, and A ∩ B has 3 elements, how many elements are in A

a) 12

b) 9

c) 15

d) 8

Therefore, the contrapositive of "If it's sunny, then I'll go to the beach" is "If I don't go to the beach,

15. In set theory, what does A × B represent?

a) The union of A and B

b) The intersection of A and B

c) The Cartesian product of A and B

d) The set difference of A and B

Answer: c) The Cartesian product of A and B

Solution: A × B represents the Cartesian product of sets A and B. It is the set of all ordered pairs (a,

b) where a is an element of A and b is an element of B.

16. Which of the following is equivalent to the statement "Not all birds can fly"?

a) All birds cannot fly

c) Some birds cannot fly

Solution: The negation of "All A are B" is "Some A are not B". Therefore, "Not all birds can fly" is

17. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is the cardinality of A ∆ B? (∆ denotes symmetric

b) 4

c) 6

d) 8

Solution: The symmetric difference A ∆ B contains elements that are in either A or B, but not in

both. Here, A ∆ B = {1, 2, 5, 6}. The cardinality (number of elements) of this set is 4.

b) A' ∪ B'

c) (A' ∩ B) ∪ (A ∩ B')

d) ∅

Answer: a) A' ∩ B'

Solution: Using De Morgan's laws:

(A ∩ B)' = A' ∪ B'

This simplifies to A' ∩ B'

19. If p: "It's raining" and q: "I'll take an umbrella", which of the following represents "I'll take an

umbrella if and only if it's raining"?

a) p → q

d) ¬p ∨ q

propositional logic. It means that the statement is true when both parts have the same truth value.

a) p ∧ ¬p

c) p → q

Answer: a) If I don't go to the beach, then it's not sunny

Solution: The contrapositive of a conditional statement "If p, then q" is "If not q, then not p".

Therefore, the contrapositive of "If it's sunny, then I'll go to the beach" is "If I don't go to the beach,

Answer: c) The Cartesian product of A and B

Solution: A × B represents the Cartesian product of sets A and B. It is the set of all ordered pairs (a,

a) All birds cannot fly

b) No birds can fly

d) Some birds can fly

Answer: c) Some birds cannot fly

d) 8

Answer: b) 4

18. What is the result of the operation (A ∪ B)' ∩ (A ∩ B)'?

d) ∅

(A ∩ B)' = A' ∪ B'

So, (A ∪ B)' ∩ (A ∩ B)' = (A' ∩ B') ∩ (A' ∪ B')

This simplifies to A' ∩ B'

a) p → q

Solution: The phrase "if and only if" is represented by the biconditional operator (↔) in

a) 15

b) 25

Answer: a) 15

Solution: Let's use the inclusion-exclusion principle:

Total = Math + Physics - Both + Neither

100 = 65 + 45 - 25 + Neither

values of its components. p ∧ ¬p (read as "p and not p") is always false, making it a contradiction.

a) 5

b) 8

Answer: b) 8

Solution: To negate a conditional statement p → q, we assert p and negate q. Therefore, ¬(p → q)

B. In a Venn diagram, this corresponds to shading only the region where the two circles overlap.

25. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is P(A ∩ B)? (P(X) denotes the power set of X)

a) {{}, {3}, {4}, {3, 4}}

- The empty set {}

- The set itself {3, 4}

. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

a) True

c) Indeterminate

3. F ∨ T = T

Therefore, the final result is True.

A ∩ B?

a) {2}

b) {2, 3, 5, 7}

c) {4, 6, 8}

Answer: a) {2}

The intersection of A and B (A ∩ B) contains elements that are in both A and B. The only such

6. Which of the following is logically equivalent to p → q?

b) p ∧ q

d) q → p

also to the disjunction ¬p ∨ q. This is a fundamental equivalence in propositional logic.

b) {2, 4, 6, 8, 10}

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

a) If the grass is wet, it rains

c) If the grass doesn't get wet, it doesn't rain

converse of "If it rains, the grass gets wet" is "If the grass is wet, it rains".

9. In a Venn diagram representing A ∪ B, which region is shaded?

a) Only the region inside circle A

b) Only the region inside circle B

c) The regions inside both circles A and B, including their overlap

d) Only the region where circles A and B overlap

Answer: c) The regions inside both circles A and B, including their overlap

Solution: The union of sets A and B (A ∪ B) includes all elements that are in A, or in B, or in both. In

10. What is the negation of the statement "x < 5 and y > 3"?

a) x ≥ 5 or y ≤ 3

c) x < 5 or y > 3

Answer: a) x ≥ 5 or y ≤ 3

Solution: To negate a compound statement connected by "and", we negate each part and change

"and" to "or". So, the negation of "x < 5 and y > 3" is "not(x < 5) or not(y > 3)", which simplifies to "x

11. If A = {1, 2, 3} and B = {3, 4, 5}, what is A - B?

a) {1, 2}

b) {4, 5}

c) {1, 2, 4, 5}

Answer: a) {1, 2}

12. Which of the following is a tautology?

c) p → q

d) p ↔ q

b) 9

c) 15

d) 8

Solution: We can use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Here, n(A ∪ B) = 5 + 7 - 3 = 9

14. Which of the following is the contrapositive of "If it's sunny, then I'll go to the beach"?

b) If it's not sunny, then I won't go to the beach

then it's not sunny".

Answer: c) The Cartesian product of A and B

16. Which of the following is equivalent to the statement "Not all birds can fly"?

d) Some birds can fly

Answer: c) Some birds cannot fly

Solution: The negation of "All A are B" is "Some A are not B". Therefore, "Not all birds can fly" is

equivalent to "Some birds cannot fly".

b) 4

c) 6

Answer: b) 4

b) A' ∪ B'

a) {1, 2}

c) {1, 2, 4, 5}

Answer: a) {1, 2}

Solution: The set difference A - B contains all elements that are in A but not in B. In this case, 1 and

2 are in A but not in B, while 3 is in both sets. Therefore, A - B = {1, 2}.

12. Which of the following is a tautology?

c) 15

Answer: b) 9

Solution: We can use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

a) If I don't go to the beach, then it's not sunny

d) If it's sunny, then I won't go to the beach

Answer: a) If I don't go to the beach, then it's not sunny

Therefore, the contrapositive of "If it's sunny, then I'll go to the beach" is "If I don't go to the beach,

then it's not sunny".

15. In set theory, what does A × B represent?

a) The union of A and B

c) The Cartesian product of A and B

d) The set difference of A and B

b) where a is an element of A and b is an element of B.

b) No birds can fly

d) Some birds can fly

Answer: c) Some birds cannot fly

Solution: The negation of "All A are B" is "Some A are not B". Therefore, "Not all birds can fly" is

equivalent to "Some birds cannot fly".

17. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is the cardinality of A ∆ B? (∆ denotes symmetric

Solution: The symmetric difference A ∆ B contains elements that are in either A or B, but not in

This simplifies to A' ∩ B'

19. If p: "It's raining" and q: "I'll take an umbrella", which of the following represents "I'll take an

b) q → p

propositional logic. It means that the statement is true when both parts have the same truth value.

b) 25

c) 35

Answer: a) 15

Solution: Let's use the inclusion-exclusion principle:

Total = Math + Physics - Both + Neither

100 = 65 + 45 - 25 + Neither

Neither = 100 - 65 - 45 + 25 = 15

21. Which of the following is a contradiction?

a) p ∧ ¬p

b) p ∨ ¬p

c) p → q

d) p ↔ q

Answer: a) p ∧ ¬p

Solution: A contradiction is a compound proposition that is always false regardless of the truth

a) 5

b) 8

c) 10

d) 3

Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

a) ¬p → ¬q

Solution: To negate a conditional statement p → q, we assert p and negate q. Therefore, ¬(p → q)

is equivalent to p ∧ ¬q.

b) Only the region inside circle B

Answer: d) Only the region where circles A and B overlap

25. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is P(A ∩ B)? (P(X) denotes the power set of X)

d) {3, 4}

Answer: a) {{}, {3}, {4}, {3, 4}}

Solution: First, A ∩ B = {3, 4}

Therefore, P(A ∩ B) = {{}, {3}, {4}, {3, 4}}

. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

a) {1, 2, 3, 4, 5}

b) {3}

or in both. Therefore, A ∪ B = {1, 2, 3, 4, 5}.

4. What is the truth value of the compound proposition: (T ∧ F) ∨ (T ∨ F)?

c) Indeterminate

Solution: Let's evaluate step by step:

5. If A = {x | x is an even number less than 10} and B = {x | x is a prime number less than 10}, what is

a) {2}

c) {4, 6, 8}

A = {2, 4, 6, 8}

The intersection of A and B (A ∩ B) contains elements that are in both A and B. The only such

element is 2.

6. Which of the following is logically equivalent to p → q?

a) ¬p ∨ q

Answer: a) ¬p ∨ q

7. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, what is A'? (A' denotes the complement of

a) {1, 3, 5, 7, 9}

c) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

8. Which of the following is the converse of "If it rains, the grass gets wet"?

b) If it doesn't rain, the grass doesn't get wet

c) If the grass doesn't get wet, it doesn't rain

d) It rains if and only if the grass gets wet

Answer: a) If the grass is wet, it rains

9. In a Venn diagram representing A ∪ B, which region is shaded?

a) Only the region inside circle A

d) Only the region where circles A and B overlap

Answer: c) The regions inside both circles A and B, including their overlap

a Venn diagram, this corresponds to shading all regions inside either circle.

10. What is the negation of the statement "x < 5 and y > 3"?

c) x < 5 or y > 3

"and" to "or". So, the negation of "x < 5 and y > 3" is "not(x < 5) or not(y > 3)", which simplifies to "x

c) {1, 2, 4, 5}

b) p ∨ ¬p

c) p → q

13. If A has 5 elements and B has 7 elements, and A ∩ B has 3 elements, how many elements are in A

b) 9

d) 8

Answer: b) 9

Here, n(A ∪ B) = 5 + 7 - 3 = 9

14. Which of the following is the contrapositive of "If it's sunny, then I'll go to the beach"?

Therefore, the contrapositive of "If it's sunny, then I'll go to the beach" is "If I don't go to the beach,

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

b) If it doesn't rain, the grass doesn't get wet

c) If the grass doesn't get wet, it doesn't rain

d) It rains if and only if the grass gets wet

Answer: a) If the grass is wet, it rains

9. In a Venn diagram representing A ∪ B, which region is shaded?

a) Only the region inside circle A

b) Only the region inside circle B

c) The regions inside both circles A and B, including their overlap

d) Only the region where circles A and B overlap

Answer: c) The regions inside both circles A and B, including their overlap

a Venn diagram, this corresponds to shading all regions inside either circle.

10. What is the negation of the statement "x < 5 and y > 3"?

a) x ≥ 5 or y ≤ 3

b) x ≥ 5 and y ≤ 3

d) x ≥ 5 and y > 3

Answer: a) x ≥ 5 or y ≤ 3

c) {1, 2, 4, 5}

d) {3}

Answer: a) {1, 2}

Solution: The set difference A - B contains all elements that are in A but not in B. In this case, 1 and

12. Which of the following is a tautology?

c) p → q

of its components. p ∨ ¬p (read as "p or not p") is always true, making it a tautology.

a) 12

b) 9

d) 8

Answer: a) If I don't go to the beach, then it's not sunny

then it's not sunny".

15. In set theory, what does A × B represent?

a) The union of A and B

b) The intersection of A and B

c) The Cartesian product of A and B

b) where a is an element of A and b is an element of B.

16. Which of the following is equivalent to the statement "Not all birds can fly"?

b) No birds can fly

equivalent to "Some birds cannot fly".

17. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is the cardinality of A ∆ B? (∆ denotes symmetric

difference)

a) 2

b) 4

d) 8

Answer: b) 4

Solution: The symmetric difference A ∆ B contains elements that are in either A or B, but not in

both. Here, A ∆ B = {1, 2, 5, 6}. The cardinality (number of elements) of this set is 4.

b) A' ∪ B'

Answer: a) A' ∩ B'

(A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

c) p ↔ q

Answer: c) p ↔ q

Solution: The phrase "if and only if" is represented by the biconditional operator (↔) in

Answer: a) 15

Solution: Let's use the inclusion-exclusion principle:

100 = 65 + 45 - 25 + Neither

Neither = 100 - 65 - 45 + 25 = 15

a) p ∧ ¬p

b) p ∨ ¬p

d) p ↔ q

a) 5

b) 8

c) 10

Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

The cardinality of this set is 8.

23. Which of the following is logically equivalent to ¬(p → q)?

a) ¬p → ¬q

b) p ∧ ¬q

c) ¬p ∨ q

d) q → p

Answer: b) p ∧ ¬q

Solution: To negate a conditional statement p → q, we assert p and negate q. Therefore, ¬(p → q)

is equivalent to p ∧ ¬q.

24. In a Venn diagram representing A ∩ B, which region is shaded?

b) Only the region inside circle B

B. In a Venn diagram, this corresponds to shading only the region where the two circles overlap.

25. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is P(A ∩ B)? (P(X) denotes the power set of X)

a) {{}, {3}, {4}, {3, 4}}

Answer: a) {{}, {3}, {4}, {3, 4}}

Solution: First, A ∩ B = {3, 4}

The power set of {3, 4} includes:

- Single element sets {3} and {4}

- The set itself {3, 4}

c) {1, 2, 4, 5}

Answer: a) {1, 2, 3, 4, 5}

d) Neither true nor false

Answer: a) True

Solution: Let's evaluate step by step:

1. (T ∧ F) = F

2. (T ∨ F) = T

3. F ∨ T = T

Therefore, the final result is True.

5. If A = {x | x is an even number less than 10} and B = {x | x is a prime number less than 10}, what is

A ∩ B?

a) {2}

B = {2, 3, 5, 7}

The intersection of A and B (A ∩ B) contains elements that are in both A and B. The only such

element is 2.

6. Which of the following is logically equivalent to p → q?

a) ¬p ∨ q

b) p ∧ q

c) ¬q → ¬p

d) q → p

Answer: a) ¬p ∨ q

Solution: The conditional statement p → q is logically equivalent to its contrapositive ¬q → ¬p, and

also to the disjunction ¬p ∨ q. This is a fundamental equivalence in propositional logic.

7. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, what is A'? (A' denotes the complement of

A)

not in A. Therefore, A' = {1, 3, 5, 7, 9}.

8. Which of the following is the converse of "If it rains, the grass gets wet"?

a) If the grass is wet, it rains

b) If it doesn't rain, the grass doesn't get wet

c) If the grass doesn't get wet, it doesn't rain

d) It rains if and only if the grass gets wet

Answer: a) If the grass is wet, it rains

Solution: The converse of a conditional statement "If p, then q" is "If q, then p". Therefore, the

converse of "If it rains, the grass gets wet" is "If the grass is wet, it rains".

9. In a Venn diagram representing A ∪ B, which region is shaded?

a) Only the region inside circle A

b) Only the region inside circle B

a) x ≥ 5 or y ≤ 3

b) x ≥ 5 and y ≤ 3

c) x < 5 or y > 3

d) x ≥ 5 and y > 3

Answer: a) x ≥ 5 or y ≤ 3

Solution: To negate a compound statement connected by "and", we negate each part and change

"and" to "or". So, the negation of "x < 5 and y > 3" is "not(x < 5) or not(y > 3)", which simplifies to "x

≥ 5 or y ≤ 3".

11. If A = {1, 2, 3} and B = {3, 4, 5}, what is A - B?

a) {1, 2}

Answer: a) True

Solution: Let's evaluate step by step:

A ∩ B?

b) {2, 3, 5, 7}

Solution: The conditional statement p → q is logically equivalent to its contrapositive ¬q → ¬p, and

Answer: c) The regions inside both circles A and B, including their overlap

10. What is the negation of the statement "x < 5 and y > 3"?

a) x ≥ 5 or y ≤ 3

b) x ≥ 5 and y ≤ 3

a) p ∧ ¬p

d) p ↔ q

of its components. p ∨ ¬p (read as "p or not p") is always true, making it a tautology.

13. If A has 5 elements and B has 7 elements, and A ∩ B has 3 elements, how many elements are in A

Solution: We can use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

b) If it's not sunny, then I won't go to the beach

d) If it's sunny, then I won't go to the beach

Answer: a) If I don't go to the beach, then it's not sunny

Solution: The contrapositive of a conditional statement "If p, then q" is "If not q, then not p".

Therefore, the contrapositive of "If it's sunny, then I'll go to the beach" is "If I don't go to the beach,

then it's not sunny".

15. In set theory, what does A × B represent?

a) The union of A and B

c) The Cartesian product of A and B

Answer: c) The Cartesian product of A and B

Solution: A × B represents the Cartesian product of sets A and B. It is the set of all ordered pairs (a,

b) where a is an element of A and b is an element of B.

16. Which of the following is equivalent to the statement "Not all birds can fly"?

a) All birds cannot fly

b) No birds can fly

difference)

a) 2

d) 8

Answer: b) 4

Solution: The symmetric difference A ∆ B contains elements that are in either A or B, but not in

both. Here, A ∆ B = {1, 2, 5, 6}. The cardinality (number of elements) of this set is 4.

18. What is the result of the operation (A ∪ B)' ∩ (A ∩ B)'?

b) A' ∪ B'

c) (A' ∩ B) ∪ (A ∩ B')

d) ∅

Answer: a) A' ∩ B'

Solution: Using De Morgan's laws:

a) p → q

b) q → p

c) p ↔ q

d) ¬p ∨ q

Answer: c) p ↔ q

Solution: The phrase "if and only if" is represented by the biconditional operator (↔) in

propositional logic. It means that the statement is true when both parts have the same truth value.

20. In a group of 100 students, 65 study mathematics, 45 study physics, and 25 study both. How

many students study neither mathematics nor physics?

a) 15

d) 45

Answer: a) 15

Total = Math + Physics - Both + Neither

Neither = 100 - 65 - 45 + 25 = 15

Answer: b) 8

Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

The cardinality of this set is 8.

23. Which of the following is logically equivalent to ¬(p → q)?

c) ¬p ∨ q

is equivalent to p ∧ ¬q.

24. In a Venn diagram representing A ∩ B, which region is shaded?

a) Only the region inside circle A

c) The regions inside both circles A and B, including their overlap

B. In a Venn diagram, this corresponds to shading only the region where the two circles overlap.

25. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is P(A ∩ B)? (P(X) denotes the power set of X)

b) {{3}, {4}, {3, 4}}

c) {{}, {3, 4}}

d) {3, 4}

- The empty set {}

- The set itself {3, 4}

26. Which of the following is the negation of "For all x, if x > 0 then x^2 > 0"?

b) There exists an x such that x > 0 and x^2 ≤ 0