Logical Equivalence

Logical Equivalence

Given the propositions P and Q, the logical equivalence of P and Q is the new proposition noted P ⇔ Q, that is true if and only if the biconditional P ↔︎ Q is a tautology.

Symbolism

  • The logical equivalence of the propositions P and Q is noted as “P ⇔ Q” which is read as “P is logically equivalent to Q”.
  • This true table shows that the propositions P and Q are equivalent.
P Q P → Q ¬P ¬P ∨ Q (P → Q) ↔ (¬P ∨ Q)
T T T F T T
T F F F F T
F T T T T T
F F T T T T

Example

Consider the propositions “P: 15 is a multiple of a” and “Q: 15 is divisible by a” where a ∈ {1,3,5,15}.
The propositions P and Q are equivalent because they have the same solution set.

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