Union of disjoint sets
High school
Consider two sets that have no elements in common. Their union is a union of disjoint sets.
Undergraduate
Consider sets $A$ and $B$ with $A \cap B = \emptyset$. 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
- $A = \lbrace 1, 2 \rbrace$ and $B = \lbrace 3, 4 \rbrace$ : $A \cup B = \lbrace 1, 2, 3, 4 \rbrace$ Elementary
-
$A = \lbrace \text{apple} \rbrace$ and $B = \lbrace \text{banana} \rbrace$
: $A \cup B = \lbrace \text{apple}, \text{banana} \rbrace$
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.
-
$A = \lbrace \text{apple}, \text{orange} \rbrace$ and $B = \lbrace \text{orange}, \text{banana} \rbrace$
: $A \cup B$ is not a disjoint union because $A$ and $B$ overlap
High school
$A$ and $B$ overlap, so their ordinary union $\lbrace \text{apple}, \text{orange}, \text{banana} \rbrace$ is not a disjoint union in either sense. The tagged disjoint union keeps both copies of orange, so it has $4$ elements.
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.