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.