Logical equivalence

In logic and mathematics, two statements are logically equivalent if they can prove each other (under a set of axioms),^[1] or have the same truth value under all circumstances. In propositional logic, two statements are logically equivalent precisely when their truth tables are identical.^[2] To express logical equivalence between two statements, the symbols ${\displaystyle \equiv }$, ${\displaystyle \Leftrightarrow }$ and ${\displaystyle \iff }$are often used.^[3][4]

For example, the statements "A and B" and "B and A" are logically equivalent.^[2] If P and Q are logically equivalent, then the statement "P if and only if Q" is a tautology.^[4]

1. ↑ "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-10-09.
2. ↑ ^{2.0 2.1} "Section 1.1: Logical Forms and Equivalencies". www.csm.ornl.gov. Retrieved 2020-10-09.
3. ↑ "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-10-09.
4. ↑ ^{4.0 4.1} "2.5: Logical Equivalences". Mathematics LibreTexts. 2019-08-13. Retrieved 2020-10-09.