# Definition:Disjoint Sets

## 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.

$S$ 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.