ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D77
Set union

Let $E_j$ be a D11: Set for every $j \in J$.
The union of $E = \{ E_j \}_{j \in J}$ is the D11: Set $$\bigcup_{j \in J} E_j : = \{ x \mid \exists \, j \in J : x \in E_j \}$$

Let $x$ be a D11: Set.
The union of $x$ is the D11: Set $$\cup x : = \{ z \mid \exists \, y \in x : z \in y \}$$
Children
 ▶ D157: Disjoint union ▶ D74: Set cover ▶ D589: Successor set
Results
 ▶ R2223: Binary set union is commutative ▶ R4145: Binary union is an upper bound to both sets in the union ▶ R4213: Countable set union is invariant under bijective shifting of indices ▶ R4143: Countable union is an upper bound to each set in the union ▶ R4214: Finite set union is invariant under bijective shifting of indices ▶ R4144: Finite union is an upper bound to each set in the union ▶ R2071: Isotonicity of set union ▶ R4167: Superset of binary union iff superset of both sets in the union ▶ R4165: Superset of countable union iff superset of every set in the union ▶ R4166: Superset of finite union iff superset of every set in the union ▶ R292: Union is smallest upper bound ▶ R2080: Union is upper bound for intersection