TeXipedia

lneq

Represents a strict less-than relation that is specifically not equal, combining the concepts of 'less than' and 'not equal' into a single symbol.

Overview

Serves as a specialized mathematical comparison operator commonly used in advanced mathematics and formal proofs where precise inequality relationships need to be expressed.

  • Essential in set theory and analysis for expressing strict ordering relationships
  • Often appears in mathematical logic and abstract algebra when standard inequalities need refinement
  • Particularly useful when distinguishing between strict and non-strict orderings in formal mathematical writing
  • Provides a more concise alternative to writing separate 'less than' and 'not equal' conditions

Examples

Comparing two real numbers where one is strictly less than but not equal to the other.

x⪇y  ⟹  x<y and x≠yx \lneq y \implies x < y \text{ and } x \neq y 
x \lneq y \implies x < y \text{ and } x \neq y

Showing strict subset relationship between two sets.

A⪇B  ⟺  A⊂B and A≠BA \lneq B \iff A \subset B \text{ and } A \neq B 
A \lneq B \iff A \subset B \text{ and } A \neq B

Demonstrating strict inequality in a sequence convergence.

an⪇L for all n∈Na_n \lneq L \text{ for all } n \in \mathbb{N} 
a_n \lneq L \text{ for all } n \in \mathbb{N}