Logical Equivalence Quiz: Identify and Prove Equivalent Statements

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.

In propositional logic, P ∧ Q represents the logical requirement that both P and Q must simultaneously be true. If even one fails, the conjunction no longer holds because the strength of the AND operator demands total agreement. Thus, a single false input is enough to make the entire expression false.

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.

The expression P ∨ P does not increase the range of situations where the compound statement is true; P already covers all such possibilities. Adding another copy of P does not provide any additional scenarios because OR returns true whenever at least one copy of P is true—and since both copies are identical, the expression still evaluates exactly like P alone. This is why P ∨ P simplifies directly to P.

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

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

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.”

A biconditional means both implications must hold: If P then Q, and if Q then P. Only when both directions work do P and Q have the same truth value.

Q ∧ ¬Q is a contradiction, always false.

So the expression becomes:

P ∨ False = P.

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

Contrapositives reverse and negate both sides:

P → Q is equivalent to ¬Q → ¬P.

They always share the same truth value.

This is the absorption law. If P is true, the whole expression is true regardless of Q. If P is false, both sides are false. Thus the expression collapses simply to P.

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

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

P(Q + R) ↔ P ∧ (Q ∨ R).

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

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.

De Morgan’s law states that ¬(P ∨ Q) is logically equivalent to (¬P ∧ ¬Q). This happens because if the entire disjunction P ∨ Q is false, then neither P nor Q can be true; both must fail simultaneously. The negation therefore forces a conjunction of the negated parts, capturing the idea that “neither P nor Q is true.

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.

The implication P → Q is defined as false in exactly one situation: when P is true and Q is false. In all other truth assignments, it is considered true. The expression ¬P ∨ Q is true precisely in those same situations—it is false only when ¬P is false (meaning P is true) and Q is false. Because both expressions share the same truth table, they are logically equivalent.