Union of disjoint sets
High school
Consider two sets $A$ and $B$ that have no elements in common. The union $A \cup B$ is a union of disjoint sets.
Undergraduate
Consider sets $A$ and $B$ such that $A \cap B = \emptyset$. A union of disjoint sets is the ordinary union $A \cup B$ in this case. Writing $A \sqcup B$ (or $A \mathbin{\dot{\cup}} B$) for it asserts that $A$ and $B$ are disjoint.
Examples
Worked examples
- $A = \setdef{1, 2}$ and $B = \setdef{3, 4}$ : $A \cup B = \setdef{1, 2, 3, 4}$ Elementary
-
$A = \setdef{\text{apple}}$ and $B = \setdef{\text{banana}}$
: $A \cup B = \setdef{\text{apple}, \text{banana}}$
High school
$A$ and $B$ are disjoint, so their union is a union of disjoint sets. The tagged disjoint union is usually identified with it.
-
$A = \setdef{\text{apple}, \text{orange}}$ and $B = \setdef{\text{orange}, \text{banana}}$
: None
High school
The union of $A = \setdef{\text{apple}, \text{orange}}$ and $B = \setdef{\text{orange}, \text{banana}}$ is not a union of disjoint sets, because $A$ and $B$ overlap.
Distinctions involving this term
- Disjoint union: union of disjoint sets vs. tagged union: “Disjoint union” can mean an ordinary union of sets that happen to be disjoint, or a construction that tags each element with the set it came from, so that copies of overlapping sets are kept apart.