Logical Equivalence Quiz: Identify and Prove Equivalent Statements

Double negation cancels itself out. Saying “it is not the case that P is false” is exactly the same as simply stating P.

In the expression P ∨ (Q ∧ P), the presence of P ensures the entire statement becomes true whenever P is true. The subexpression Q ∧ P cannot ever introduce a new true case beyond those already covered by P, because Q ∧ P can only be true when P is already true. Thus, the entire expression simplifies to just P; it does not expand the set of situations where the statement is true.

This is the Idempotent Law of Conjunction, which states that repeating the same logical condition in an AND statement does not change its truth value.



If P is true, then P ∧ P is true ∧ true, which is true.



If P is false, then P ∧ P is false ∧ false, which is false.



Since the truth value matches P in every case, the entire expression reduces simply to P.

The second occurrence of P adds no new information.

Factor out P using distributive law, similar to factoring in algebra:

P(Q + R) ↔ P ∧ (Q ∨ R).

Contrapositives reverse and negate both sides:

P → Q is equivalent to ¬Q → ¬P.

They always share the same truth value.

This is De Morgan’s Law. Negating an AND flips it into an OR and negates each component. The statement “not both P and Q” is equivalent to “either P is false or Q is false.”

This is the Law of Excluded Middle. Every proposition is either true or not true — one must hold — making the statement a tautology.

Q ∧ ¬Q is a contradiction, always false.

So the expression becomes:

P ∨ False = P.

The expression P → Q is equivalent to ¬P ∨ Q. Now consider the behavior of ¬P ∨ Q: if P is true, ¬P is false, leaving the value determined by Q. But in the larger implication context, P being true and Q being true satisfies the implication, and when P is false, ¬P becomes true, making the entire statement true regardless of Q.

Negating an OR turns it into an AND (De Morgan). The incorrect OR form was rightly marked false.

¬P → ¬Q is the contrapositive of Q → P. A conditional is always logically equivalent to its contrapositive, so Q → P is the correct match.

The biconditional P ↔ Q asserts that P and Q are logically equivalent—they are either both true or both false. Negating this expression gives ¬(P ↔ Q), which flips the criterion: it becomes true when P and Q no longer match. This is exactly the definition of exclusive OR, where the compound statement is true only when one is true and the other is false. It enforces the requirement that P and Q have opposite truth values.

Modus ponens is one of the foundational valid argument forms in logic. If we know that P → Q is true (meaning Q must be true whenever P is true), and we are additionally told that P is indeed true, then Q becomes an unavoidable conclusion. The implication guarantees that Q holds in all cases where P is satisfied.

¬P → ¬Q is the contrapositive of Q → P. Contrapositives always preserve truth.

A biconditional means both directions of implication must hold simultaneously. P↔Q requires that if P then Q and also if Q then P. Only when both conditional directions are true do P and Q share the same truth value in all cases. Option A uses OR which does not require both directions. Options C and D do not capture bidirectionality.

This is the absorption law. If P is true the whole expression is true regardless of Q since P∨Q is also true. If P is false then P∧(P∨Q) is false regardless of Q. The truth value always matches P exactly, so the expression collapses to P. Option A expands rather than simplifies. Option B incorrectly introduces a conjunction with Q.

De Morgan's law states that ¬(P∨Q) = ¬P∧¬Q. If the entire disjunction is false, neither P nor Q can be true — both must be false simultaneously. The negation forces a conjunction of the negated parts. Option A is De Morgan applied to ¬(P∧Q) not ¬(P∨Q). Options C and D remove the negation entirely.

By De Morgan's law, negating a disjunction produces a conjunction of the negations: ¬(P∨Q) = ¬P∧¬Q. If the entire OR is false then neither component can be true, forcing both to be false simultaneously. Option A applies the wrong De Morgan form. Options B and D remove negations incorrectly.

The conjunction P∧Q requires both components to be true simultaneously. If either P or Q is false the entire conjunction is false. Option A requires only P. Option B requires only Q. Option C describes disjunction not conjunction. Only option D correctly states the AND condition.

By the idempotent law of disjunction, P∨P = P. Adding a second copy of P introduces no new truth scenarios since OR is already true whenever P is true. Option A is the law of excluded middle, a tautology. Option B is conjunction not disjunction. Option C negates P, giving the opposite truth value.