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Leftrightarrow

Represents logical or mathematical equivalence, indicating that two statements imply each other or are mutually equivalent.

Overview

Essential in mathematical proofs, logical reasoning, and formal mathematics where bidirectional implications need to be shown. This double-headed arrow serves as a fundamental symbol in:

  • Mathematical logic for expressing biconditional statements ("if and only if")
  • Set theory for showing set equivalence
  • Algebraic proofs where expressions are shown to be identical
  • Computer science and discrete mathematics for logical equivalences

Commonly used in textbooks, academic papers, and formal mathematical writing where precise logical relationships need to be expressed.

Examples

Logical equivalence in a mathematical statement showing two expressions are equal if and only if.

x2=4⇔x=±2x^2 = 4 \Leftrightarrow x = \pm 2 
x^2 = 4 \Leftrightarrow x = \pm 2

Showing the equivalence of two set theory expressions.

A⊆B and B⊆A⇔A=BA \subseteq B \text{ and } B \subseteq A \Leftrightarrow A = B 
A \subseteq B \text{ and } B \subseteq A \Leftrightarrow A = B

Demonstrating logical biconditional in a mathematical definition.

n is even⇔∃k∈Z:n=2k\text{$n$ is even} \Leftrightarrow \exists k \in \mathbb{Z} : n = 2k 
\text{$n$ is even} \Leftrightarrow \exists k \in \mathbb{Z} : n = 2k