Definition:Set Intersection/Finite Intersection
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Definition
Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$.
Then:
- $\ds S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$
where $\N^*_n = \set {1, 2, 3, \ldots, n}$.
If it is clear from the context that $i \in \N^*_n$, we can also write $\ds \bigcap_{\N^*_n} S_i$.
Also defined as
The specific nature of the indexing set $\N^*_n$ is immaterial; some treatments may use a zero-based set, thus:
- $\ds S = \bigcap_{i \mathop \in \N_n} S_i = \set {x: \forall i \in \N_n: x \in S_i}$
where $\N_n = \set {0, 1, 2, \ldots, n - 1}$.
In this context the sets under consideration are $S = S_0 \cap S_1 \cap \ldots \cap S_{n - 1}$.
The distinction is sufficiently trivial as to be hardly worth mentioning.
Also denoted as
Other notations for this concept are:
- $\ds \bigcap_{i \mathop = 1}^n S_i$
- $\ds \bigcap_{1 \mathop \le i \mathop \le n} S_i$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 1$: Operations on Sets
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.8$. Sets of sets
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 5$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7$: Unions and Intersections
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): intersection