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.