# Definition:Set Intersection/Family of Sets

## Contents

## 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:

- $\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

### 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:

- $\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in \mathbb U: \forall i \in I: x \in S_i}$

### 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:

- $\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$

## 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 $S_i$:

- $\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

it follows that:

- $\displaystyle \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$

## Also denoted as

The set $\displaystyle \bigcap_{i \mathop \in I} S_i$ can also be seen denoted as:

- $\displaystyle \bigcap_I S_i$

or, if the indexing set is clear from context:

- $\displaystyle \bigcap_i S_i$

However, on this website it is recommended that the full form is used.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.2$: 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 - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.2$: 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