Definition:Disjoint Sets/Family
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Definition
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
Then $\family {S_i}_{i \mathop \in I}$ is disjoint if and only if their intersection is empty:
- $\ds \bigcap_{i \mathop \in I} S_i = \O$
Examples
$3$ Arbitrary Sets
Let $I = \set {1, 2, 3}$ be an indexing set.
Let:
\(\ds S_1\) | \(=\) | \(\ds \set {a, b}\) | ||||||||||||
\(\ds S_2\) | \(=\) | \(\ds \set {b, c}\) | ||||||||||||
\(\ds S_3\) | \(=\) | \(\ds \set {a, c}\) |
Then the family of sets $\family {S_i}_{i \mathop \in I}$ is disjoint, but not pairwise disjoint.
Also see
Sources
- 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets