Definition:Disjoint Sets

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Definition

Two sets $S$ and $T$ are disjoint if and only if:

$S \cap T = \O$

That is, disjoint sets are sets such that their intersection is the empty set -- they have no elements in common.


Euler Diagram

The concept of disjoint sets can be illustrated in the following Euler diagram.

EulerDiagram-DisjointSets.png

$S$ and $T$ are disjoint.


Family

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:

$\ds \bigcap_{i \mathop \in I} S_i = \O$


Also known as

Some early sources refer to disjoint sets as non-overlapping or non-intersecting.

The term mutually exclusive sets can also be seen.


Examples

Arbitrary Example

The sets $\set {1, 2}$ and $\set {4, 5}$ are disjoint.


$3$ Arbitrarily Chosen Sets

Let:

\(\ds U\) \(=\) \(\ds \set {u_1, u_2, u_3}\)
\(\ds V\) \(=\) \(\ds \set {u_1, u_3}\)
\(\ds W\) \(=\) \(\ds \set {u_2, u_4}\)


Then:

\(\ds U \cup V\) \(=\) \(\ds \set {u_1, u_2, u_3}\)
\(\ds U \cap V\) \(=\) \(\ds \set {u_1, u_3}\)
\(\ds V \cup W\) \(=\) \(\ds \set {u_1, u_2, u_3, u_4}\)
\(\ds U \cap W\) \(=\) \(\ds \set {u_2}\)
\(\ds V \cap W\) \(=\) \(\ds \O\)

Thus $V$ and $W$ are disjoint.


Also see

  • Results about disjoint sets can be found here.


Sources

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