# Subset Relation

(Redirected from Subset)

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

## 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 $\displaystyle{ A \subseteq B }$ ,

:or equivalently

:* B is a superset of (or includes) A, denoted by $\displaystyle{ B \supseteq A. }$ 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 $\displaystyle{ A \subsetneq B. }$ :or equivalently

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

When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.