TeXipedia

subsetneq

Denotes a strict subset relationship where one set is properly contained within another, with explicit inequality.

Overview

Commonly used in set theory and mathematical proofs to explicitly indicate that one set is a proper subset of another, emphasizing that the sets are not equal.

  • Essential for precise mathematical notation when distinguishing between proper and improper subsets
  • Frequently appears in abstract algebra, topology, and analysis
  • Particularly useful when constructing rigorous mathematical arguments that rely on strict set containment
  • Often used alongside other set relation symbols to express complex set-theoretic relationships

Examples

Showing a proper subset relationship between sets of numbers.

N⊊Z⊊Q⊊R\mathbb{N} \subsetneq \mathbb{Z} \subsetneq \mathbb{Q} \subsetneq \mathbb{R} 
\mathbb{N} \subsetneq \mathbb{Z} \subsetneq \mathbb{Q} \subsetneq \mathbb{R}

Demonstrating proper subset relationship between finite sets.

{1,2}⊊{1,2,3}\{1,2\} \subsetneq \{1,2,3\} 
\{1,2\} \subsetneq \{1,2,3\}

Expressing proper subset relationship in set builder notation.

{x∈R:x2<4}⊊{x∈R:x2≤4}\{x \in \mathbb{R} : x^2 < 4\} \subsetneq \{x \in \mathbb{R} : x^2 \leq 4\} 
\{x \in \mathbb{R} : x^2 < 4\} \subsetneq \{x \in \mathbb{R} : x^2 \leq 4\}