Set Difference with Union
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Theorem
Let $R, S, T$ be sets.
Then:
- $R \setminus \paren {S \cup T} = \paren {R \cup T} \setminus \paren {S \cup T} = \paren {R \setminus S} \setminus T = \paren {R \setminus T} \setminus S$
where:
- $R \setminus S$ denotes set difference
- $R \cup T$ denotes set union.
Illustration by Venn Diagram
Proof
Consider $R, S, T \subseteq \mathbb U$, where $\mathbb U$ is considered as the universe.
\(\ds \paren {R \cup T} \setminus \paren {S \cup T}\) | \(=\) | \(\ds \paren {R \cup T} \cap \overline {\paren {S \cup T} }\) | Set Difference as Intersection with Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \cup T} \cap \paren {\overline S \cap \overline T}\) | De Morgan's Laws: Complement of Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {R \cup T} \cap \overline T} \cap \overline S\) | Intersection is Associative and Intersection is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {R \cup T} \setminus T} \setminus S\) | Set Difference as Intersection with Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \setminus T} \setminus S\) | Set Difference with Union is Set Difference |
$\Box$
Then:
\(\ds R \setminus \paren {S \cup T}\) | \(=\) | \(\ds R \cap \overline {\paren {S \cup T} }\) | Set Difference as Intersection with Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds R \cap \paren {\overline S \cap \overline T}\) | De Morgan's Laws: Complement of Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \cap \overline S} \cap \overline T\) | Intersection is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \setminus S} \setminus T\) | Set Difference as Intersection with Complement |
$\Box$
Then:
\(\ds R \setminus \paren {S \cup T}\) | \(=\) | \(\ds R \setminus \paren {T \cup S}\) | Union is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \setminus T} \setminus S\) | from above |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(c)}$
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$: Problem $1 \ \text{(ii)}$