Disjoint Sets¶

Definition¶

Two sets $A$ and $B$ are said to be disjoint if they intersect emptily, i.e., $A\cap B =\varnothing\text{.}$

Interpretation¶

Disjoint sets share no common elements: an element of one set is ruled out of the other. The condition concerns only their intersection, not how the sets are described.

In the left panel the discs $A$ and $B$ have no common points, so $A\cap B=\varnothing$$\varnothing$. In the right panel the shaded overlap contains points of both sets, so the sets are not disjoint. A single common element is enough to break disjointness, however small the overlap.

Disjointness makes counting additive: for disjoint finite sets, $|A\cup B|=|A|+|B|$$|B|$, because no element is counted twice. A set and its complement are always disjoint, and so are $A\setminus B$ and $B$. For more than two sets, "pairwise disjoint" means that every two of them are disjoint; this is stronger than an empty common intersection, as $\{1,2\}$, $\{2,3\}$ and $\{1,3\}$ show.

This notion formalizes the idea of mutual exclusivity in probability: events that cannot happen at the same time.

Key Results¶

• If $A$ and $B$ are disjoint finite sets, then $|A\cup B|=|A|+|B|\text{.}$
• Disjoint sets are distinct, with one exception: $\varnothing\cap\varnothing=\varnothing\text{,}$ so the empty set is disjoint from itself. It is the only such set, because $A\cap A=A\text{.}$

Examples¶

• The sets $\{1,\,2\}$ and $\{3,\,4\}$ are disjoint.
• Any set $A$ and $\varnothing$ are disjoint.

Prerequisites¶

• Intersection of Sets
• Empty Set

Related Concepts¶

• Difference Set
• Complement
• Disjoint Events

Sources¶

• Math I: Calculus · Lecture 1

Symbols: intersection, union

Last updated: 20 October 2026 03:02