Definition:Set Intersection/Family of Sets
Definition
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
Then the intersection of $\family {S_i}$ is defined as:
- $\ds \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$
In the context of the Universal Set
In treatments of set theory in which the concept of the universal set is recognised, this can be expressed as follows.
Let $\mathbb U$ be a universal set.
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$.
Then the intersection of $\family {S_i}$ is defined and denoted as:
- $\ds \bigcap_{i \mathop \in I} S_i := \set {x \in \mathbb U: \forall i \in I: x \in S_i}$
Subsets of General Set
This definition is the same when the universal set $\mathbb U$ is replaced by any set $X$, which may or may not be a universal set:
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$.
Then the intersection of $\family {S_i}$ is defined as:
- $\ds \bigcap_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$
where $i$ is a bound variable.
Intersection of Family of Two Sets
Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
From the definition of the intersection of $S_i$:
- $\ds \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$
it follows that:
- $\ds \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$
Also denoted as
The set $\ds \bigcap_{i \mathop \in I} S_i$ can also be seen denoted as:
- $\ds \bigcap_I S_i$
or, if the indexing set is clear from context:
- $\ds \bigcap_i S_i$
Some sources use the form:
- $\bigcap \set {S_i: i \in I}$
Examples
Example: $\size {y - 1} < n$ and $\size {y + 1} > \dfrac 1 n$
Let $I$ be the indexing set $I = \set {1, 2, 3, \ldots}$
Let $\family {T_n}$ be the indexed family of subsets of the set of real numbers $\R$, defined as:
- $T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$
Then:
- $\ds \bigcap_{n \mathop \in I} T_n = \openint 0 2$
Also see
- Results about set intersections can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 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
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7$: Unions and Intersections
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets: Exercise $1.4.3: \ \text{(ii)}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.2$: Boolean Operations
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $12$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products