Question 1:

What is the truth value of the logical statement "¬(P ∧ Q)" if P is true and Q is false?

Explanation: The logical statement "¬(P ∧ Q)" is the negation of the conjunction of P and Q. Since P is true and Q is false, the conjunction "P ∧ Q" is false. Therefore, the negation "¬(P ∧ Q)" is true.

Question 2:

What is the truth value of the logical statement "P ∨ (Q ∧ R)" if P is true, Q is false, and R is true?

Explanation: The logical statement "P ∨ (Q ∧ R)" is the disjunction of P and the conjunction of Q and R. Since P is true, the disjunction "P ∨ (Q ∧ R)" is true regardless of the truth values of Q and R.

Question 3:

Which of the following truth tables represents the logical operator "exclusive disjunction" (XOR)?

P | Q | P XOR Q
--------------
T | T |   F
T | F |   T
F | T |   T
F | F |   F

P | Q | P XOR Q
--------------
T | T |   T
T | F |   F
F | T |   F
F | F |   T

P | Q | P XOR Q
--------------
T | T |   T
T | F |   T
F | T |   T
F | F |   F

P | Q | P XOR Q
--------------
T | T |   F
T | F |   T
F | T |   F
F | F |   T

Explanation: The truth table that represents the logical operator "exclusive disjunction" (XOR) is option B. In XOR, the result is true if the inputs have different truth values and false if the inputs have the same truth value.

Question 4:

Which of the following logical deductions is valid?

Explanation: The logical deduction that is valid is option A, Modus Ponens. Modus Ponens states that if we have a conditional statement "If P, then Q" and we know P is true, then we can conclude that Q is true.

Question 5:

What is the truth value of the logical statement "¬(P → Q) ∧ R" if P is false, Q is true, and R is true?

Explanation: The logical statement "¬(P → Q) ∧ R" is the conjunction of the negation of the conditional statement "P → Q" and R. Since P is false and Q is true, the conditional "P → Q" is false. Therefore, the negation "¬(P → Q)" is true. Additionally, since R is true, the conjunction "¬(P → Q) ∧ R" is true.