TeXipedia

Longleftarrow

Represents a long double arrow pointing leftward, commonly used in mathematical logic and formal proofs to indicate logical implication or reverse mapping.

Overview

Serves as a prominent notation in advanced mathematics and formal reasoning, particularly when emphasizing important logical relationships or mappings.

  • Essential in mathematical proofs and logical arguments to show reverse implications
  • Used in category theory to denote special types of morphisms or functors
  • Appears in set theory and abstract algebra for depicting reverse mappings
  • Often paired with \Longrightarrow to show logical equivalence or bidirectional relationships
  • Particularly useful in formal mathematical writing where visual distinction between different types of arrows is important

Examples

Logical implication written in reverse, showing the left-hand side follows from the right.

x=2⟸x2=4 and x>0x = 2 \Longleftarrow x^2 = 4 \text{ and } x > 0 
x = 2 \Longleftarrow x^2 = 4 \text{ and } x > 0

Showing a necessary condition in mathematical reasoning.

Matrix is singular⟸det⁡(A)=0\text{Matrix is singular} \Longleftarrow \det(A) = 0 
\text{Matrix is singular} \Longleftarrow \det(A) = 0

Indicating reverse logical sequence in a mathematical proof.

f(x)=0⟸x−1=0⟸x=1f(x) = 0 \Longleftarrow x - 1 = 0 \Longleftarrow x = 1 
f(x) = 0 \Longleftarrow x - 1 = 0 \Longleftarrow x = 1