Definition:Disjoint Sets

Definition

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.

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

$3$ Arbitrarily Chosen Sets

Let:

 $\displaystyle U$ $=$ $\displaystyle \set {u_1, u_2, u_3}$ $\quad$ $\quad$ $\displaystyle V$ $=$ $\displaystyle \set {u_1, u_3}$ $\quad$ $\quad$ $\displaystyle W$ $=$ $\displaystyle \set {u_2, u_4}$ $\quad$ $\quad$

Then:

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

Thus $V$ and $W$ are disjoint.