Disjoint union
Undergraduate
Consider two sets $A$ and $B$, which may overlap. Their disjoint union is a set that combines them while keeping their elements apart: $A \sqcup B = (\lbrace 1 \rbrace \times A) \cup (\lbrace 2 \rbrace \times B)$, so each element is tagged with the set it came from.
Examples
Worked examples
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$A = \lbrace \text{apple} \rbrace$ and $B = \lbrace \text{banana} \rbrace$
: $A \sqcup B = \lbrace (1, \text{apple}), (2, \text{banana}) \rbrace$
High school
Each element is tagged with $1$ (from $A$) or $2$ (from $B$). The result is usually identified with $\lbrace \text{apple}, \text{banana} \rbrace$.
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$A = \lbrace \text{apple}, \text{orange} \rbrace$ and $B = \lbrace \text{orange}, \text{banana} \rbrace$
: $A \sqcup B = \lbrace (1, \text{apple}), (1, \text{orange}), (2, \text{orange}), (2, \text{banana}) \rbrace$
High school
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.