# Subset Relation

A Subset Relation is a binary set relation between sets ([math]\displaystyle{ X_1,X_2 }[/math]) that is True If [math]\displaystyle{ \forall x \in X_1: x \in X_2 }[/math].

**AKA:**SubsetOf Operation, Inclusion Relation, ⊆.**Example(s):**- {} ⊆ {1, 2, 3}.
- {1, 2, 3} ⊆ {1, 2, 3}.
- {1, 2} ⊆ {1, 2, 3}, also a Proper Subset Relation(⊂)
- a Subclass Relation.
- a Population Subset.
- …

**Counter-Example(s):**- a Not Subset Relation (⊄), such as
`{A} ⊄ {1, 2, 3}`

. - Superset.

- a Not Subset Relation (⊄), such as
**See:**Set of Subsets, Empty Set.

## References

### 2017

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Subset#Definitions Retrieved:2017-6-8.
- If
*A*and*B*are sets and every element of*A*is also an element of*B*, then::*

*A*is a**subset**of (or is included in)*B*, denoted by [math]\displaystyle{ A \subseteq B }[/math] ,:or equivalently

:*

*B*is a**superset**of (or includes)*A*, denoted by [math]\displaystyle{ B \supseteq A. }[/math] If*A*is a subset of*B*, but*A*is not equal to*B*(i.e. there exists at least one element of B which is not an element of*A*), then:*

*A*is also a**proper**(or**strict**)**subset**of*B*; this is written as [math]\displaystyle{ A \subsetneq B. }[/math] :or equivalently:*

*B*is a**proper superset**of*A*; this is written as [math]\displaystyle{ B \supsetneq A. }[/math] For any set*S*, the inclusion relation ⊆ is a partial order on the set [math]\displaystyle{ \mathcal{P}(S) }[/math] of all subsets of*S*(the power set of*S*) defined by [math]\displaystyle{ A \leq B \iff A \subseteq B }[/math] . We may also partially order [math]\displaystyle{ \mathcal{P}(S) }[/math] by reverse set inclusion by defining [math]\displaystyle{ A \leq B \iff B \subseteq A }[/math] .When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.

- If

### 2009

- (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
- a set whose members are members of another set; a set contained within another set