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Union

The set of elements that belong to at least one of the given sets.
Union

The union of sets AA and BB is

A∪B:={x:(x∈A) ∨ (x∈B)}.A\cup B := \{x : (x\in A)\ \lor\ (x\in B)\}.

More generally, for an indexed family {Ai}i∈I\{A_i\}_{i\in I}, the union is

⋃i∈IAi:={x:∃i∈I with x∈Ai}.\bigcup_{i\in I} A_i := \{x : \exists i\in I\ \text{with}\ x\in A_i\}.

Unions are central in topology and analysis: open sets are closed under arbitrary unions, and coverings are families whose union contains the set of interest.

Examples:

  • {1,2}∪{2,3}={1,2,3}\{1,2\}\cup\{2,3\}=\{1,2,3\}.
  • (0,1)∪(1,2)=(0,2)∖{1}(0,1)\cup(1,2)=(0,2)\setminus\{1\}.
  • If An:=(−1/n,1/n)⊆RA_n:=(-1/n,1/n)\subseteq\mathbb{R}, then ⋃n=1∞An=(−1,1)\bigcup_{n=1}^\infty A_n = (-1,1).