Distributive Laws/Set Theory
Theorem
Intersection Distributes over Union
Set intersection is distributive over set union:
- $R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$
Union Distributes over Intersection
Set union is distributive over set intersection:
- $R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$
Examples
Example: $A \cap B \cap \paren {C \cup D} \subseteq \paren {A \cap D} \cup \paren {B \cap C}$
Let:
- $P = A \cap B \cap \paren {C \cup D}$
- $Q = \paren {A \cap D} \cup \paren {B \cap C}$
Then:
- $P \subseteq Q$
Also defined as
Some sources include in the distributive laws (of set theory) other identities, for example:
Symmetric Difference Distributes over Intersection
Symmetric difference is distributive over intersection:
| \(\ds R \symdif \paren {S \cap T}\) | \(=\) | \(\ds \paren {R \symdif S} \cap \paren {R \symdif T}\) |
Also known as
The distributive laws (of set theory) are also known as:
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $2 \ \text{(d)}$
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1961: John G. Hocking and Gail S. Young: Topology ... (previous) ... (next): A Note on Set-Theoretic Concepts: $(3)$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.3$: Set operations
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.2$: Operations on sets: $(1)$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 7 \ \text{(a)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.4 \ \text{(i)}$: Unions and Intersections
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distributive law
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distributive
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets: $\text {(vi)}$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(vi)}$