TeXipedia

Updownarrow

Represents a double-stroke vertical bidirectional arrow commonly used in mathematical logic and set theory.

Overview

Serves as a logical symbol in formal mathematics and theoretical computer science, particularly in contexts involving bidirectional implications or equivalences.

  • Frequently used to denote logical biconditionals or if-and-only-if relationships
  • Appears in proof systems and formal reasoning
  • Common in set theory when showing mutual containment or equivalence between sets
  • Distinguished from the single-stroke variant by its more emphatic appearance, often indicating stronger relationships

Examples

Showing logical equivalence between two mathematical statements.

p∧q⇕¬(¬p∨¬q)p \land q \Updownarrow \neg(\neg p \lor \neg q) 
p \land q \Updownarrow \neg(\neg p \lor \neg q)

Indicating bidirectional implication in a system of equations.

x>0⇕x2>0x > 0 \Updownarrow x^2 > 0 
x > 0 \Updownarrow x^2 > 0

Representing mutual dependence between mathematical expressions.

det⁡(A)=0⇕A is singular\det(A) = 0 \Updownarrow A \text{ is singular} 
\det(A) = 0 \Updownarrow A \text{ is singular}