De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union/Proof 2
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Theorem
- $\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$
Proof
Let $x \in S$ througout.
\(\ds \) | \(\) | \(\ds x \in \relcomp S {T_1 \cup T_2}\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \notin \paren {T_1 \cup T_2}\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \neg \paren {x \in T_1 \lor x \in T_2}\) | Definition of Set Union | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \notin T_1 \land x \notin T_2\) | De Morgan's Laws: Conjunction of Negations | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \in \relcomp S {T_1} \land x \in \relcomp S {T_2}\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \in \relcomp S {T_1} \cap \relcomp S {T_2}\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \relcomp S {T_1 \cup T_2} \subseteq \relcomp S {T_1} \cap \relcomp S {T_2}\) | Definition of Subset |
\(\ds \) | \(\) | \(\ds x \in \relcomp S {T_1} \cap \relcomp S {T_2}\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \in \relcomp S {T_1} \land x \in \relcomp S {T_2}\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \notin T_1 \land x \notin T_2\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \neg \paren {x \in T_1 \lor x \in T_2}\) | De Morgan's Laws: Conjunction of Negations | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \notin \paren {T_1 \cup T_2}\) | Definition of Set Union | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x \in \relcomp S {T_1 \cup T_2}\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \relcomp S {T_1} \cap \relcomp S {T_2} \subseteq \relcomp S {T_1 \cup T_2}\) | Definition of Set Intersection |
By definition of set equality:
- $\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Theorem $3.2$