De Morgan's Laws (Set Theory)/Set Difference/Difference with Union

Theorem
Let $S, T_1, T_2$ be sets.

Then:
 * $S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$

where:
 * $T_1 \cap T_2$ denotes set intersection
 * $T_1 \cup T_2$ denotes set union.


 * DeMorganMinusUnion.png

Proof
By definition of set equality:
 * $S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$