Subset Relation

Jump to navigation Jump to search

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].



  • (Wikipedia, 2017) ⇒ 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}.