ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Collection of sets
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
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