Transforming Statements Using Equivalence Laws Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) Which pair of statements is logically equivalent?

Explanation

The conditional statement P → Q is logically equivalent to its contrapositive ¬Q → ¬P.

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About This Quiz
Transforming Statements Using Equivalence Laws Quiz - Quiz

Think you can recognize when a complicated statement is secretly just a simpler one in disguise? This quiz focuses on using distributive, absorption, and De Morgan’s laws—as well as contrapositives and biconditionals—to transform and simplify propositional formulas. You’ll rewrite everyday language into symbolic form, apply rules step by step, and... see moredecide which expressions match each other logically. As you progress, you’ll sharpen your ability to move between different but equivalent forms, a key skill for proofs and deeper logical reasoning. see less

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2) Which expression is logically equivalent to (P ∧ Q) ∨ (P ∧ R)?

Explanation

This is the distributive law of AND over OR. (P ∧ Q) ∨ (P ∧ R) means either (P and Q) or (P and R). This is equivalent to P and (Q or R).

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3) ¬(P ∨ Q) is logically equivalent to:

Explanation

This is De Morgan's Law: the negation of a disjunction is the conjunction of the negations.
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4) Which is logically equivalent to the statement 'It is not true that both Alice and Bob failed'?

Explanation

The statement is equivalent to ¬(A ∧ B), where A is "Alice failed" and B is "Bob failed". By De Morgan's Law, this is equivalent to ¬A ∨ ¬B, which means either Alice passed or Bob passed.
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5) Which expression is logically equivalent to (P → Q) ∧ P?

Explanation

We have the following chain of logical equivalences: (P → Q) ∧ P ≡ (¬P ∨ Q) ∧ P ≡ (¬P ∧ P) ∨ (Q ∧ P) ≡ F ∨ (Q ∧ P) ≡  P ∧ Q. This means if P is true and P implies Q, then Q must also be true.

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6) The statement P ∨ (P ∧ Q) is logically equivalent to:

Explanation

P ∨ (P ∧ Q) ≡ P. This is the absorption law. If P is true, then the whole expression is true regardless of Q.

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7) Which statement is logically equivalent to 'You can't be both happy and sad'?

Explanation

The statement is equivalent to ¬(H ∧ S), where H is "happy" and S is "sad". By De Morgan's Law, this is equivalent to ¬H ∨ ¬S, which means if you're happy (H is true), then you're not sad (¬S is true).
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8) Which expression is logically equivalent to ¬P → ¬Q?

Explanation

¬P → ¬Q is the contrapositive of Q → P. The contrapositive of any implication is logically equivalent to the original implication.

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9) Which pair is logically equivalent?

Explanation

P → Q is equivalent to ¬P ∨ Q. This is the material implication equivalence.
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10) ¬(P ↔ Q) is logically equivalent to:

Explanation

The negation of P ↔ Q (P if and only if Q) is equivalent to P ⊕ Q (P exclusive or Q), which means P and Q have different truth  values. This is also equivalent to P ↔ ¬Q and ¬P ↔ Q.
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11) Which statement is logically equivalent to ¬(P ∧ Q)?

Explanation

This is De Morgan's Law for the negation of AND. The negation of "P and Q" is equivalent to "not P OR not Q".

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12) P ∧ (Q ∨ ¬Q) is equivalent to:

Explanation

Q ∨ ¬Q is always true (tautology). Therefore, P ∧ (Q ∨ ¬Q) is equivalent to P ∧ true, which is simply P.
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13) Which is logically equivalent to P ∨ (Q ∧ R)?

Explanation

This is the distributive Law of OR over AND: P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R).
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14) Which expression is logically equivalent to P → (Q → R)?

Explanation

By the exportation law, P → (Q → R) is logically equivalent to (P ∧ Q) → R.
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15) (P ∧ Q) ∧ R is equivalent to:

Explanation

This is the associative law for conjunction. Since all the connectives are AND operations, the grouping of the AND operations does not affect the result.
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Which pair of statements is logically equivalent?
Which expression is logically equivalent to (P ∧ Q) ∨ (P ∧ R)?
¬(P ∨ Q) is logically equivalent to:
Which is logically equivalent to the statement 'It is not true that...
Which expression is logically equivalent to (P → Q) ∧ P?
The statement P ∨ (P ∧ Q) is logically equivalent to:
Which statement is logically equivalent to 'You can't be both happy...
Which expression is logically equivalent to ¬P → ¬Q?
Which pair is logically equivalent?
¬(P ↔ Q) is logically equivalent to:
Which statement is logically equivalent to ¬(P ∧ Q)?
P ∧ (Q ∨ ¬Q) is equivalent to:
Which is logically equivalent to P ∨ (Q ∧ R)?
Which expression is logically equivalent to P → (Q → R)?
(P ∧ Q) ∧ R is equivalent to:
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