TeXipedia

nRightarrow

Represents a negated double right arrow, indicating that a logical implication or mapping does not hold.

Overview

Essential in mathematical logic and set theory for expressing non-implications or the absence of specific relationships between statements or sets.

  • Commonly used in proof writing to show counter-examples or disprove implications.
  • Appears frequently in abstract algebra and topology to demonstrate non-existence of certain mappings.
  • Particularly useful in advanced mathematics where precise logical negation is required.
  • Often paired with its positive counterpart (\Rightarrow) to contrast valid and invalid implications.

Examples

Showing that two statements are not logically equivalent.

A⇏BA \nRightarrow B 
A \nRightarrow B

Demonstrating non-implication in set theory.

x∈A⇏x∈Bx \in A \nRightarrow x \in B 
x \in A \nRightarrow x \in B

Expressing that convergence in one metric does not imply convergence in another.

∥xn−x∥→0⇏d(xn,x)→0\|x_n - x\| \to 0 \nRightarrow d(x_n,x) \to 0 
\|x_n - x\| \to 0 \nRightarrow d(x_n,x) \to 0