Simplifying Propositions With Classical Logic Laws Quiz

1) P ∧ Q is equivalent to:

¬P ∧ ¬Q

P ∨ Q

Q ∧ P

P → Q

This is the commutative law for conjunction. The order of the operands in an AND operation does not affect the result.

Explanation

This is the commutative law for conjunction. The order of the operands in an AND operation does not affect the result.

2) P ∨ Q is equivalent to:

¬P ∨ ¬Q

P ∧ Q

Q ∨ P

Q → P

This is the commutative law for disjunction. The order of the operands in an OR operation does not affect the result.

Explanation

This is the commutative law for disjunction. The order of the operands in an OR operation does not affect the result.

3) Which form matches ¬P ∨ Q?

P → Q

P ∧ Q

¬(P ∧ Q)

¬P → ¬Q

¬P ∨ Q is the material implication equivalence of P → Q. We can think of it as "if P then Q" can be rephrased as "either P is not true, or Q is true."

Explanation

¬P ∨ Q is the material implication equivalence of P → Q. We can think of it as "if P then Q" can be rephrased as "either P is not true, or Q is true."

4) ¬(P ∧ Q) is equivalent to:

¬P ∧ ¬Q

¬P ∨ ¬Q

P ∨ Q

P ∧ Q

This is De Morgan's Law applied to the negation of a conjunction. The expression ¬(P ∧ Q) means 'it is not the case that both P and Q are true,' which is equivalent to saying 'either P is not true, or Q is not true,' expressed as ¬P ∨ ¬Q.

Explanation

This is De Morgan's Law applied to the negation of a conjunction. The expression ¬(P ∧ Q) means 'it is not the case that both P and Q are true,' which is equivalent to saying 'either P is not true, or Q is not true,' expressed as ¬P ∨ ¬Q.

5) Simplify ¬(P ∨ Q)

¬P ∧ ¬Q

¬P ∨ ¬Q

P ∧ Q

P ∨ Q

This is De Morgan's Law applied to the negation of a disjunction. The expression ¬(P ∨ Q) means 'it is not the case that P or Q is true,' which is equivalent to saying 'P is not true and Q is not true,' expressed as ¬P ∧ ¬Q.

Explanation

This is De Morgan's Law applied to the negation of a disjunction. The expression ¬(P ∨ Q) means 'it is not the case that P or Q is true,' which is equivalent to saying 'P is not true and Q is not true,' expressed as ¬P ∧ ¬Q.

6) The contrapositive of P → Q is:

¬Q → ¬P

Q → P

¬P → ¬Q

P → ¬Q

The contrapositive of the implication P → Q is ¬Q → ¬P. This means that 'if P then Q' is logically equivalent to 'if not Q then not P.'

Explanation

The contrapositive of the implication P → Q is ¬Q → ¬P. This means that 'if P then Q' is logically equivalent to 'if not Q then not P.'

7) P ∧ (P ∨ Q) is equivalent to:

P

Q

P ∨ Q

P ∧ Q

The expression P ∧ (P ∨ Q) means 'P is true and either P is true or Q is true.' If P is true, the entire statement is true regardless of Q. Thus, the expression simplifies to just P.

Explanation

The expression P ∧ (P ∨ Q) means 'P is true and either P is true or Q is true.' If P is true, the entire statement is true regardless of Q. Thus, the expression simplifies to just P.

8) P ∧ P is equivalent to:

P ∨ P

¬P

P ∧ ¬P

P

The expression P ∧ P means 'P is true and P is true.' This is redundant and simplifies to just P. This is known as the Idempotent Law for conjunction.

Explanation

The expression P ∧ P means 'P is true and P is true.' This is redundant and simplifies to just P. This is known as the Idempotent Law for conjunction.

9) Which law states that P ∨ (Q ∧ P) ≡ P?

Commutative law

Associative law

Distributive law

Absorption law

The expression P ∨ (Q ∧ P) means 'P is true or both Q and P are true.' If P is true, the entire statement is true regardless of Q. Thus, the expression simplifies to just P. This is known as the Absorption Law.

Explanation

The expression P ∨ (Q ∧ P) means 'P is true or both Q and P are true.' If P is true, the entire statement is true regardless of Q. Thus, the expression simplifies to just P. This is known as the Absorption Law.

10) P ∨ (¬P ∧ Q) is logically equivalent to:

P ∨ Q

P ∧ Q

¬P ∨ Q

P → Q

The expression P ∨ (¬P ∧ Q) can be simplified using the distributive property: (P ∨ ¬P) ∧ (P ∨ Q). Since P ∨ ¬P is always true, the expression simplifies to True ∧ (P ∨ Q), which is equivalent to P ∨ Q.

Explanation

The expression P ∨ (¬P ∧ Q) can be simplified using the distributive property: (P ∨ ¬P) ∧ (P ∨ Q). Since P ∨ ¬P is always true, the expression simplifies to True ∧ (P ∨ Q), which is equivalent to P ∨ Q.

11) Which is equivalent to ¬(¬P)?

P

¬P

P ∧ P

P ∨ P

The negation of the negation of P is equivalent to P. If P is true, then ¬P is false, and ¬(¬P) is true. If P is false, then ¬P is true, and ¬(¬P) is false.

Explanation

The negation of the negation of P is equivalent to P. If P is true, then ¬P is false, and ¬(¬P) is true. If P is false, then ¬P is true, and ¬(¬P) is false.

12) P ↔ Q is equivalent to:

P ∧ Q

(P → Q) ∨ (Q → P)

(P → Q) ∧ (Q → P)

¬P ∨ Q

The biconditional statement P ↔ Q means 'P is true if and only if Q is true.' This is equivalent to saying 'if P is true then Q is true, and if Q is true then P is true,' which is expressed as (P → Q) ∧ (Q → P).

Explanation

The biconditional statement P ↔ Q means 'P is true if and only if Q is true.' This is equivalent to saying 'if P is true then Q is true, and if Q is true then P is true,' which is expressed as (P → Q) ∧ (Q → P).

13) Which statement is a tautology (always true)?

P ∧ ¬P

P ∨ ¬P

P → ¬P

P ↔ ¬P

The expression P ∨ ¬P means 'P is true or P is not true.' This is always true because P must be either true or false. This is known as the Law of Excluded Middle.

Explanation

The expression P ∨ ¬P means 'P is true or P is not true.' This is always true because P must be either true or false. This is known as the Law of Excluded Middle.