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

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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

\(\displaystyle \) \(\) \(\displaystyle x \in S \setminus \paren {T_1 \cup T_2}\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S} \land \paren {x \notin \paren {T_1 \cup T_2} }\) Definition of Set Difference
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S} \land \paren {\neg \paren {x \in T_1 \lor x \in T_2} }\) Definition of Set Union
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S} \land \paren {x \notin T_1 \land x \notin T_2}\) De Morgan's Laws: Conjunction of Negations
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S \land x \notin T_1} \land \paren {x \in S \land x \notin T_2}\) Rules of Idempotence, Commutation and Association
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle x \in \paren {S \setminus T_1} \cap \paren {S \setminus T_2}\) Definition of Set Intersection and Definition of Set Difference

By definition of set equality:

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

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources