TeXipedia

Longleftrightarrow

Represents a logical bi-implication or material equivalence in mathematical proofs and logical statements.

Overview

Serves as a formal symbol for "if and only if" relationships in mathematical logic, proofs, and theoretical computer science.

  • Essential in formal mathematical definitions where two statements are shown to be logically equivalent
  • Common in set theory, abstract algebra, and formal logic
  • Often used to connect equivalent mathematical expressions or conditions
  • Distinguished from the shorter double arrow by emphasizing important logical equivalences or major theorem statements

Examples

Logical equivalence in a mathematical statement showing two conditions are equivalent.

x2=4⟺x=±2x^2 = 4 \Longleftrightarrow x = \pm 2 
x^2 = 4 \Longleftrightarrow x = \pm 2

Showing the equivalence of two mathematical definitions or expressions.

f is continuous at a⟺lim⁡x→af(x)=f(a)f \text{ is continuous at } a \Longleftrightarrow \lim_{x \to a} f(x) = f(a) 
f \text{ is continuous at } a \Longleftrightarrow \lim_{x \to a} f(x) = f(a)

Demonstrating logical biconditional in set theory.

x∈A∩B⟺x∈A and x∈Bx \in A \cap B \Longleftrightarrow x \in A \text{ and } x \in B 
x \in A \cap B \Longleftrightarrow x \in A \text{ and } x \in B