TeXipedia

Leftarrow

Represents a double-lined left arrow symbol commonly used in mathematical logic and set theory to indicate logical implication or mapping relationships.

Overview

Essential in formal mathematics and theoretical computer science for expressing logical relationships and mappings between sets or concepts.

  • Frequently used to denote reverse implications in logical proofs
  • Common in category theory to represent morphisms
  • Appears in set theory to show mappings or correspondences
  • Distinguished from the single arrow (\leftarrow) by its stronger visual emphasis, often indicating more significant or formal relationships

Examples

Logical implication written in reverse (if and only if).

q⇐pq \Leftarrow p 
q \Leftarrow p

System of equations showing backward implications.

x>0⇐x2>0⇐x4>0x > 0 \Leftarrow x^2 > 0 \Leftarrow x^4 > 0 
x > 0 \Leftarrow x^2 > 0 \Leftarrow x^4 > 0

Mathematical definition with necessary condition.

prime(p)⇐p has exactly two factors\text{prime}(p) \Leftarrow p \text{ has exactly two factors} 
\text{prime}(p) \Leftarrow p \text{ has exactly two factors}