Subset Relation

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

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 $A \subseteq B$ ,

:or equivalently

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

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

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