TeXipedia

nleftrightarrow

Represents a negated bidirectional arrow indicating logical non-equivalence or the absence of a two-way relationship between expressions.

Overview

Essential in mathematical logic and set theory for denoting non-equivalence or the absence of mutual implication between statements or expressions.

  • Commonly used in formal proofs to show that two statements are not logically equivalent
  • Appears in abstract algebra to indicate non-isomorphism between structures
  • Useful in computer science for expressing boolean logic and truth tables
  • Often employed alongside other logical symbols to build complex mathematical arguments

Examples

Showing logical non-equivalence between two statements P and Q.

P↮QP \nleftrightarrow Q 
P \nleftrightarrow Q

Expressing that two mathematical expressions are not bidirectionally related.

x2+1>0↮x>0x^2 + 1 > 0 \nleftrightarrow x > 0 
x^2 + 1 > 0 \nleftrightarrow x > 0

Demonstrating non-biconditional relationship in set theory.

A⊆B↮B⊆AA \subseteq B \nleftrightarrow B \subseteq A 
A \subseteq B \nleftrightarrow B \subseteq A