Subset Relation

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



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] A \subseteq B [/math] ,

       :or equivalently

       :* B is a superset of (or includes) A, denoted by [math] 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] A \subsetneq B. [/math] :or equivalently

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

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

2009