Multiple Choice Questions
Advanced Topics in Mathematical Logic
Topic: Advanced Topics in Mathematical Logic
Grade: 12
Question 1:
Which of the following logical statements is equivalent to the statement \”If it is raining, then the ground is wet\”?
a) If the ground is wet, then it is raining
b) If it is not raining, then the ground is not wet
c) If the ground is not wet, then it is not raining
d) If it is raining, then the ground is not wet
Answer: c) If the ground is not wet, then it is not raining
Explanation: This is an example of the contrapositive of the original statement. The contrapositive of \”If A, then B\” is \”If not B, then not A\”. In this case, A represents \”it is raining\” and B represents \”the ground is wet\”. The contrapositive is logically equivalent to the original statement, so the correct answer is c).
Example: Simple – If it is sunny, then it is not raining. Complex – If an object is a triangle, then it has three sides.
Question 2:
Which of the following logical statements is logically equivalent to the statement \”If P then Q\”?
a) If not P, then not Q
b) If Q, then P
c) If P, then not Q
d) If not Q, then not P
Answer: a) If not P, then not Q
Explanation: The logical statement \”If P then Q\” can be rewritten as \”If not Q, then not P\”. This is known as the contrapositive of the original statement, and it is logically equivalent. So the correct answer is a).
Example: Simple – If it is a cat, then it has fur. Complex – If a number is divisible by 3, then it is divisible by 9.
Question 3:
Which of the following logical statements is logically equivalent to the statement \”P or Q\”?
a) P and Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: b) not P or not Q
Explanation: The logical statement \”P or Q\” can be rewritten as \”not P or not Q\”. This is known as De Morgan\’s law, which states that the negation of a disjunction is the conjunction of the negations. So the correct answer is b).
Example: Simple – It is raining or it is sunny. Complex – The number is divisible by 2 or it is divisible by 3.
Question 4:
Which of the following logical statements is logically equivalent to the statement \”P and Q\”?
a) P or Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: a) P or Q
Explanation: The logical statement \”P and Q\” can be rewritten as \”not (not P or not Q)\”. This is known as De Morgan\’s law, which states that the negation of a conjunction is the disjunction of the negations. So the correct answer is a).
Example: Simple – It is raining and it is sunny. Complex – The number is divisible by 2 and it is divisible by 3.
Question 5:
Which of the following logical statements is logically equivalent to the statement \”P implies Q\”?
a) P and Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: b) not P or not Q
Explanation: The logical statement \”P implies Q\” can be rewritten as \”not P or Q\”. This is known as the implication law, which states that the implication is equivalent to the disjunction of the negation of the antecedent and the consequent. So the correct answer is b).
Example: Simple – If it is a cat, then it has fur. Complex – If a number is divisible by 2, then it is not divisible by 3.
Question 6:
Which of the following logical statements is logically equivalent to the statement \”not (P and Q)\”?
a) not P or not Q
b) P or Q
c) P and Q
d) not P and not Q
Answer: a) not P or not Q
Explanation: The logical statement \”not (P and Q)\” can be rewritten as \”not P or not Q\”. This is known as De Morgan\’s law, which states that the negation of a conjunction is the disjunction of the negations. So the correct answer is a).
Example: Simple – It is not raining and it is not sunny. Complex – The number is not divisible by 2 or it is not divisible by 3.
Question 7:
Which of the following logical statements is logically equivalent to the statement \”not (P or Q)\”?
a) not P and not Q
b) P or Q
c) P and Q
d) not P or not Q
Answer: a) not P and not Q
Explanation: The logical statement \”not (P or Q)\” can be rewritten as \”not P and not Q\”. This is known as De Morgan\’s law, which states that the negation of a disjunction is the conjunction of the negations. So the correct answer is a).
Example: Simple – It is not raining or it is not sunny. Complex – The number is not divisible by 2 and it is not divisible by 3.
Question 8:
Which of the following logical statements is logically equivalent to the statement \”P if and only if Q\”?
a) P implies Q
b) Q implies P
c) P and Q
d) P or Q
Answer: c) P and Q
Explanation: The logical statement \”P if and only if Q\” can be rewritten as \”P and Q\”. This is known as the biconditional statement, which means that both P and Q are true or both P and Q are false. So the correct answer is c).
Example: Simple – It is a cat if and only if it has fur. Complex – The number is divisible by 2 if and only if it is divisible by 3.
Question 9:
Which of the following logical statements is logically equivalent to the statement \”not (P if and only if Q)\”?
a) P if and only if Q
b) not P if and only if not Q
c) P and not Q
d) not P or Q
Answer: b) not P if and only if not Q
Explanation: The logical statement \”not (P if and only if Q)\” can be rewritten as \”not P if and only if not Q\”. This is the negation of the biconditional statement, which means that either P and Q are not both true or P and Q are not both false. So the correct answer is b).
Example: Simple – It is not a cat if and only if it does not have fur. Complex – The number is not divisible by 2 if and only if it is not divisible by 3.
Question 10:
Which of the following logical statements is logically equivalent to the statement \”P xor Q\”?
a) P and Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: b) not P or not Q
Explanation: The logical statement \”P xor Q\” can be rewritten as \”not P or not Q\”. This is known as the exclusive or, which means that either P is true or Q is true, but not both. So the correct answer is b).
Example: Simple – It is sunny or it is not sunny. Complex – The number is divisible by 2 or it is not divisible by 3.
Question 11:
Which of the following logical statements is logically equivalent to the statement \”P nand Q\”?
a) P and Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: c) not P and not Q
Explanation: The logical statement \”P nand Q\” can be rewritten as \”not P and not Q\”. This is known as the negation of the conjunction, which means that both P and Q are false. So the correct answer is c).
Example: Simple – It is not sunny and it is not raining. Complex – The number is not divisible by 2 and it is not divisible by 3.
Question 12:
Which of the following logical statements is logically equivalent to the statement \”P nor Q\”?
a) P and Q
b) not P or not Q
c) not P and not Q
d) not P and Q
Answer: c) not P and not Q
Explanation: The logical statement \”P nor Q\” can be rewritten as \”not P and not Q\”. This is known as the negation of the disjunction, which means that both P and Q are false. So the correct answer is c).
Example: Simple – It is not sunny or it is not raining. Complex – The number is not divisible by 2 and it is not divisible by 3.
Question 13:
Which of the following logical statements is logically equivalent to the statement \”P implies (Q and R)\”?
a) (P implies Q) and (P implies R)
b) (P and Q) implies R
c) (P or Q) implies R
d) (P and R) implies Q
Answer: a) (P implies Q) and (P implies R)
Explanation: The logical statement \”P implies (Q and R)\” can be rewritten as \”(P implies Q) and (P implies R)\”. This is known as the distributive law of implication, which states that the implication of a conjunction is equivalent to the conjunction of the implications. So the correct answer is a).
Example: Simple – If it is sunny, then it is hot and humid. Complex – If a number is divisible by 2, then it is divisible by 3 and divisible by 4.
Question 14:
Which of the following logical statements is logically equivalent to the statement \”P or (Q and R)\”?
a) (P or Q) and (P or R)
b) (P and Q) or R
c) (P or Q) implies R
d) (P and R) implies Q
Answer: a) (P or Q) and (P or R)
Explanation: The logical statement \”P or (Q and R)\” can be rewritten as \”(P or Q) and (P or R)\”. This is known as the distributive law of disjunction, which states that the disjunction of a conjunction is equivalent to the conjunction of the disjunctions. So the correct answer is a).
Example: Simple – It is sunny or it is hot and humid. Complex – The number is divisible by 2 or it is divisible by 3 and divisible by 4.
Question 15:
Which of the following logical statements is logically equivalent to the statement \”P and (Q or R)\”?
a) (P and Q) or (P and R)
b) (P or Q) and R
c) (P and Q) implies R
d) (P or R) implies Q
Answer: a) (P and Q) or (P and R)
Explanation: The logical statement \”P and (Q or R)\” can be rewritten as \”(P and Q) or (P and R)\”. This is known as the distributive law of conjunction, which states that the conjunction of a disjunction is equivalent to the disjunction of the conjunctions. So the correct answer is a).
Example: Simple – It is sunny and it is hot or it is sunny and it is humid. Complex – The number is divisible by 2 and it is divisible by 3 or it is divisible by 2 and it is divisible by 4.