longleftrightarrow

Represents a bidirectional logical equivalence or double implication in mathematical proofs and logical statements.

Overview

Serves as an extended horizontal double-headed arrow commonly used in mathematical logic, set theory, and formal proofs to indicate that two statements are logically equivalent or mutually imply each other.

Essential in mathematical logic for showing if-and-only-if relationships
Used in formal definitions where concepts are equivalent
Appears frequently in set theory to show bijective mappings
Common in theoretical computer science for showing logical equivalences

Examples

Showing logical equivalence between two mathematical statements.

p \land q \longleftrightarrow q \land p

p \land q \longleftrightarrow q \land p

Indicating a bidirectional mapping between sets.

f: X \longleftrightarrow Y

f: X \longleftrightarrow Y

Expressing if and only if relationship in a mathematical definition.

x \text{ is even} \longleftrightarrow x = 2k \text{ for some } k \in \mathbb{Z}

x \text{ is even} \longleftrightarrow x = 2k \text{ for some } k \in \mathbb{Z}