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

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

Then:


 * $S \setminus \left({T_1 \cap T_2}\right) = \left({S \setminus T_1}\right) \cup \left({S \setminus T_2}\right)$

where:
 * $S \setminus \left({T_1 \cup T_2}\right) = \left({S \setminus T_1}\right) \cap \left({S \setminus T_2}\right)$
 * $T_1 \cap T_2$ denotes set intersection
 * $T_1 \cup T_2$ denotes set union.


 * DeMorganMinusIntersection.png DeMorganMinusUnion.png

Proof

 * $S \setminus \left({T_1 \cap T_2}\right) = \left({S \setminus T_1}\right) \cup \left({S \setminus T_2}\right)$:

So $S \setminus \left({T_1 \cap T_2}\right) = \left({S \setminus T_1}\right) \cup \left({S \setminus T_2}\right)$.


 * $S \setminus \left({T_1 \cup T_2}\right) = \left({S \setminus T_1}\right) \cap \left({S \setminus T_2}\right)$:

So $S \setminus \left({T_1 \cup T_2}\right) = \left({S \setminus T_1}\right) \cap \left({S \setminus T_2}\right)$.