Definition:Set Intersection/Finite Intersection

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Definition

Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$.

Then:

$\displaystyle S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$

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 $\displaystyle \bigcap_{\N^*_n} S_i$.


Also defined as

The specific nature of the indexing set $\N^*_n$ is immaterial; some treatments may use a zero-based set, thus:

$\displaystyle S = \bigcap_{i \mathop \in \N_n} S_i = \set {x: \forall i \in \N_n: x \in S_i}$

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:

$\displaystyle \bigcap_{i \mathop = 1}^n S_i$
$\displaystyle \bigcap_{1 \mathop \le i \mathop \le n} S_i$


Sources