Multiple Choice Questions
Discrete Mathematics: Logic and Sets
Topic: Discrete Mathematics: Logic and Sets
Grade: 11
Question 1:
Which of the following is the negation of the statement \”All cats have tails\”?
A) Some cats have tails
B) No cats have tails
C) All cats do not have tails
D) Some cats do not have tails
Answer: D) Some cats do not have tails
Explanation: The negation of the statement \”All cats have tails\” is \”Some cats do not have tails\”. This means that there exists at least one cat that does not have a tail. For example, if we have three cats, and one of them does not have a tail, then the statement \”All cats have tails\” is false, and the negation \”Some cats do not have tails\” is true.
Question 2:
Which of the following is equivalent to the statement \”If it is raining, then the ground is wet\”?
A) It is raining and the ground is wet
B) It is not raining and the ground is wet
C) It is raining and the ground is not wet
D) It is not raining and the ground is not wet
Answer: A) It is raining and the ground is wet
Explanation: The statement \”If it is raining, then the ground is wet\” can be written as \”R → W\” using logical symbols. The only way for this statement to be true is if both it is raining (R) and the ground is wet (W). Therefore, the answer is A) It is raining and the ground is wet. For example, if it is raining and the ground is wet, then the statement \”If it is raining, then the ground is wet\” is true.
Question 3:
Which of the following is the contrapositive of the statement \”If a number is divisible by 3, then it is divisible by 6\”?
A) If a number is not divisible by 6, then it is not divisible by 3
B) If a number is divisible by 6, then it is divisible by 3
C) If a number is divisible by 3, then it is not divisible by 6
D) If a number is not divisible by 3, then it is divisible by 6
Answer: B) If a number is divisible by 6, then it is divisible by 3
Explanation: The contrapositive of the statement \”If a number is divisible by 3, then it is divisible by 6\” is \”If a number is divisible by 6, then it is divisible by 3\”. This is true because if a number is divisible by 6, then it must also be divisible by 3. For example, if we have the number 12, which is divisible by 6, then it is also divisible by 3.
Question 4:
Which of the following is the conjunction of the statements \”P: It is sunny\” and \”Q: It is hot\”?
A) It is sunny or it is hot
B) It is not sunny or it is not hot
C) It is sunny and it is hot
D) It is not sunny and it is not hot
Answer: C) It is sunny and it is hot
Explanation: The conjunction of the statements \”P: It is sunny\” and \”Q: It is hot\” is the statement \”P ∧ Q\”, which means \”It is sunny and it is hot\”. This is true only if both P and Q are true. For example, if it is sunny and it is hot, then the conjunction \”It is sunny and it is hot\” is true.
Question 5:
Which of the following is the disjunction of the statements \”P: The car is red\” and \”Q: The car is blue\”?
A) The car is red or the car is blue
B) The car is not red or the car is not blue
C) The car is red and the car is blue
D) The car is not red and the car is not blue
Answer: A) The car is red or the car is blue
Explanation: The disjunction of the statements \”P: The car is red\” and \”Q: The car is blue\” is the statement \”P ∨ Q\”, which means \”The car is red or the car is blue\”. This is true if at least one of P or Q is true. For example, if the car is red, then the disjunction \”The car is red or the car is blue\” is true.
Question 6:
Which of the following is the contrapositive of the statement \”If it is sunny, then I will go to the beach\”?
A) If it is not sunny, then I will not go to the beach
B) If I will go to the beach, then it is sunny
C) If it is sunny, then I will go to the beach
D) If I will not go to the beach, then it is not sunny
Answer: A) If it is not sunny, then I will not go to the beach
Explanation: The contrapositive of the statement \”If it is sunny, then I will go to the beach\” is \”If it is not sunny, then I will not go to the beach\”. This is true because if it is not sunny, then I will not go to the beach. For example, if it is not sunny, then I will not go to the beach.
Question 7:
Which of the following is the negation of the statement \”Some birds can fly\”?
A) All birds can fly
B) No birds can fly
C) Some birds cannot fly
D) All birds cannot fly
Answer: D) All birds cannot fly
Explanation: The negation of the statement \”Some birds can fly\” is \”All birds cannot fly\”. This means that none of the birds can fly. For example, if we have three birds, and none of them can fly, then the statement \”Some birds can fly\” is false, and the negation \”All birds cannot fly\” is true.
Question 8:
Which of the following is equivalent to the statement \”If it is not raining, then the ground is not wet\”?
A) It is raining and the ground is wet
B) It is not raining and the ground is wet
C) It is raining and the ground is not wet
D) It is not raining and the ground is not wet
Answer: D) It is not raining and the ground is not wet
Explanation: The statement \”If it is not raining, then the ground is not wet\” can be written as \”¬R → ¬W\” using logical symbols. The only way for this statement to be true is if both it is not raining (¬R) and the ground is not wet (¬W). Therefore, the answer is D) It is not raining and the ground is not wet. For example, if it is not raining and the ground is not wet, then the statement \”If it is not raining, then the ground is not wet\” is true.
Question 9:
Which of the following is the contrapositive of the statement \”If a number is divisible by 4, then it is divisible by 8\”?
A) If a number is not divisible by 8, then it is not divisible by 4
B) If a number is divisible by 8, then it is divisible by 4
C) If a number is divisible by 4, then it is not divisible by 8
D) If a number is not divisible by 4, then it is divisible by 8
Answer: B) If a number is divisible by 8, then it is divisible by 4
Explanation: The contrapositive of the statement \”If a number is divisible by 4, then it is divisible by 8\” is \”If a number is divisible by 8, then it is divisible by 4\”. This is true because if a number is divisible by 8, then it must also be divisible by 4. For example, if we have the number 16, which is divisible by 8, then it is also divisible by 4.
Question 10:
Which of the following is the conjunction of the statements \”P: It is cold\” and \”Q: It is snowing\”?
A) It is cold or it is snowing
B) It is not cold or it is not snowing
C) It is cold and it is snowing
D) It is not cold and it is not snowing
Answer: C) It is cold and it is snowing
Explanation: The conjunction of the statements \”P: It is cold\” and \”Q: It is snowing\” is the statement \”P ∧ Q\”, which means \”It is cold and it is snowing\”. This is true only if both P and Q are true. For example, if it is cold and it is snowing, then the conjunction \”It is cold and it is snowing\” is true.
Question 11:
Which of the following is the disjunction of the statements \”P: The car is old\” and \”Q: The car is new\”?
A) The car is old or the car is new
B) The car is not old or the car is not new
C) The car is old and the car is new
D) The car is not old and the car is not new
Answer: A) The car is old or the car is new
Explanation: The disjunction of the statements \”P: The car is old\” and \”Q: The car is new\” is the statement \”P ∨ Q\”, which means \”The car is old or the car is new\”. This is true if at least one of P or Q is true. For example, if the car is old, then the disjunction \”The car is old or the car is new\” is true.
Question 12:
Which of the following is the contrapositive of the statement \”If it is raining, then I will stay home\”?
A) If it is not raining, then I will go out
B) If I will stay home, then it is raining
C) If it is raining, then I will stay home
D) If I will not stay home, then it is not raining
Answer: A) If it is not raining, then I will go out
Explanation: The contrapositive of the statement \”If it is raining, then I will stay home\” is \”If it is not raining, then I will go out\”. This is true because if it is not raining, then I will go out. For example, if it is not raining, then I will go out.
Question 13:
Which of the following is the negation of the statement \”Some dogs are brown\”?
A) All dogs are brown
B) No dogs are brown
C) Some dogs are not brown
D) All dogs are not brown
Answer: B) No dogs are brown
Explanation: The negation of the statement \”Some dogs are brown\” is \”No dogs are brown\”. This means that none of the dogs are brown. For example, if we have three dogs, and none of them are brown, then the statement \”Some dogs are brown\” is false, and the negation \”No dogs are brown\” is true.
Question 14:
Which of the following is equivalent to the statement \”If it is not sunny, then it is not hot\”?
A) It is sunny and it is hot
B) It is not sunny and it is hot
C) It is sunny and it is not hot
D) It is not sunny and it is not hot
Answer: D) It is not sunny and it is not hot
Explanation: The statement \”If it is not sunny, then it is not hot\” can be written as \”¬S → ¬H\” using logical symbols. The only way for this statement to be true is if both it is not sunny (¬S) and it is not hot (¬H). Therefore, the answer is D) It is not sunny and it is not hot. For example, if it is not sunny and it is not hot, then the statement \”If it is not sunny, then it is not hot\” is true.
Question 15:
Which of the following is the contrapositive of the statement \”If a number is divisible by 2, then it is divisible by 10\”?
A) If a number is not divisible by 10, then it is not divisible by 2
B) If a number is divisible by 10, then it is divisible by 2
C) If a number is divisible by 2, then it is not divisible by 10
D) If a number is not divisible by 2, then it is divisible by 10
Answer: B) If a number is divisible by 10, then it is divisible by 2
Explanation: The contrapositive of the statement \”If a number is divisible by 2, then it is divisible by 10\” is \”If a number is divisible by 10, then it is divisible by 2\”. This is true because if a number is divisible by 10, then it must also be divisible by 2. For example, if we have the number 20, which is divisible by 10, then it is also divisible by 2.