ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4172 on D70: Set difference

Difference of set and finite union equals intersection of differences

Formulation 0

Let $X, E_1, \ldots, E_N$ each be a D11: Set.

Then \begin{equation} X \setminus \bigcup_{n = 1}^N E_n = \bigcap_{n = 1}^N (X \setminus E_n) \end{equation}

Formulation 1

Let $X, E_1, \ldots, E_N$ each be a D11: Set.

Then \begin{equation} X \setminus (E_1 \cup \cdots \cup E_N) = (X \setminus E_1) \cap \cdots \cap (X \setminus E_N) \end{equation}

Proofs

Proof 0

Let $X, E_1, \ldots, E_N$ each be a D11: Set.