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.
Illustration by Venn Diagram
Proof
\(\ds \) | \(\) | \(\ds x \in S \setminus \paren {T_1 \cup T_2}\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in S} \land \paren {x \notin \paren {T_1 \cup T_2} }\) | Definition of Set Difference | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in S} \land \paren {\neg \paren {x \in T_1 \lor x \in T_2} }\) | Definition of Set Union | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in S} \land \paren {x \notin T_1 \land x \notin T_2}\) | De Morgan's Laws: Conjunction of Negations | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in S \land x \notin T_1} \land \paren {x \in S \land x \notin T_2}\) | Rules of Idempotence, Commutation and Association | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {S \setminus T_1} \cap \paren {S \setminus T_2}\) | 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
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $2 \ \text{(e)}$
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.8$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.6$. Difference and complement: Example $21$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(a)}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \ \text{(f)}$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.6$: Set Identities and Other Set Relations: Exercise $2 \ \text{(h)}$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$: Problem $1 \ \text{(v)}$
- 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions