Definition:Disjoint Sets

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

$S \cap T = \O$

That is, disjoint sets are 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.


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:

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

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.


$3$ Arbitrarily Chosen Sets


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


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

Thus $V$ and $W$ are disjoint.

Also see