Definition:Disjoint Sets/Family

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

• Definition:Pairwise Disjoint Family

Sources

• 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets