Simplifying Propositions with Classical Logic Laws Quiz

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| Questions: 14 | Updated: Dec 1, 2025
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1) P ∧ Q is equivalent to:

Explanation

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

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About This Quiz
Simplifying Propositions With Classical Logic Laws Quiz - Quiz

Want to get comfortable simplifying logical statements quickly and accurately? In this quiz, you’ll use core equivalence laws like double negation, commutativity, associativity, absorption, and the law of excluded middle. You’ll reduce expressions, identify tautologies and contradictions, and see how complex formulas collapse into clean, minimal forms. Question by question,... see moreyou’ll build fluency in rewriting propositions so that working with logical arguments feels more intuitive and less mechanical. see less

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2) P ∨ Q is equivalent to:

Explanation

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

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3) Which form matches ¬P ∨ Q?

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."
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4) ¬(P ∧ Q) is equivalent to:

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.

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5) Simplify ¬(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.

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6) P ↔ Q is equivalent to:

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).

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7) The contrapositive of P → Q is:

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.'

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8) P ∧ (P ∨ Q) is equivalent to:

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.

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9) P ∧ P is equivalent to:

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.

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

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.

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

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.

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12) P ∨ (Q ∧ ¬Q) is equivalent to:

Explanation

Q ∧ ¬Q is always false (contradiction). Therefore, P ∨ (Q ∧ ¬Q) is equivalent to P ∨ false, which is simply P.

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

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.

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14) Which is equivalent to ¬(¬P)?

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.

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P ∧ Q is equivalent to:
P ∨ Q is equivalent to:
Which form matches ¬P ∨ Q?
¬(P ∧ Q) is equivalent to:
Simplify ¬(P ∨ Q)
P ↔ Q is equivalent to:
The contrapositive of P → Q is:
P ∧ (P ∨ Q) is equivalent to:
P ∧ P is equivalent to:
Which law states that P ∨ (Q ∧ P) ≡ P?
Which statement is a tautology (always true)?
P ∨ (Q ∧ ¬Q) is equivalent to:
P ∨ (¬P ∧ Q) is logically equivalent to:
Which is equivalent to ¬(¬P)?
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