ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D77
Set union
Formulation 0
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 \begin{equation} \bigcup_{j \in J} E_j : = \{ x \mid \exists \, j \in J : x \in E_j \} \end{equation}
Formulation 1
Let $x$ be a D11: Set.
The union of $x$ is the D11: Set \begin{equation} \cup x : = \{ z \mid \exists \, y \in x : z \in y \} \end{equation}
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
▶ R5712: Partition of a binary 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
▶ R5718: Union equals empty set iff every set in the union is empty
▶ R2080: Union is a tight upper bound for intersection
▶ R292: Union is smallest upper bound