ThmDex – An index of mathematical definitions, results, and conjectures.
Result R219 on D70: Set difference
Difference of set and intersection equals union of differences
Formulation 0
Let $X$ be a D11: Set.
Let $E_j$ be a D11: Set for each $j \in J$.
Then \begin{equation} X \setminus \bigcap_{j \in J} E_j = \bigcup_{j \in J} (X \setminus E_j) \end{equation}
Formulation 1
Let $X$ be a D11: Set.
Let $E_j$ be a D11: Set for each $j \in J$.
Then \begin{equation} \left( \bigcap_{j \in J} E_j \right)^{\complement} = \bigcup_{j \in J} E^{\complement}_j \end{equation}