TeXipedia

nLeftrightarrow

Represents a negated double-headed horizontal arrow, indicating the absence of a bidirectional relationship or logical equivalence.

Overview

Serves as a critical notation in mathematical logic, set theory, and abstract algebra to denote non-equivalence between expressions or concepts.

  • Common in proof writing to show that two statements or expressions are not logically equivalent
  • Used in advanced mathematics to indicate the absence of bijective mappings between sets
  • Appears frequently in theoretical computer science when discussing non-isomorphic structures or inequivalent algorithms
  • Particularly useful in abstract algebra for demonstrating non-homeomorphic spaces or non-isomorphic groups

Examples

Showing non-equivalence between two mathematical statements.

A⇎BA \nLeftrightarrow B 
A \nLeftrightarrow B

Demonstrating that two sets are not bijective.

f:X⇎Yf: X \nLeftrightarrow Y 
f: X \nLeftrightarrow Y

Expressing that two events are neither necessary nor sufficient conditions of each other.

P(X>0)⇎P(Y>0)P(X > 0) \nLeftrightarrow P(Y > 0) 
P(X > 0) \nLeftrightarrow P(Y > 0)