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Disjoint Union

The disjoint union of two sets A and B is a binary operator that combines all distinct elements of a pair of given sets, while retaining the original set membership as a distinguishing characteristic of the union set. For sets, the notation A⊔B is commonly used for the disjoint union (Prasolov 2006, p. xii).

The notation A union ^*B denotes the same operation. One concrete construction that keeps the copies of any common elements distinct is

A union ^*B=(A×{0}) union (B×{1})=A^* union B^*,
(1)

where A×S is a Cartesian product. For example, the disjoint union of sets A={1,2,3,4,5} and B={1,2,3,4} can be computed by finding

A^*={(1,0),(2,0),(3,0),(4,0),(5,0)}
(2)
B^*={(1,1),(2,1),(3,1),(4,1)},
(3)

so

A union ^*B=A^* union B^*
(4)
={(1,0),(2,0),(3,0),(4,0),(5,0),(1,1),(2,1),(3,1),(4,1)}.
(5)

See also

Graph Disjoint Union, Graph Union, Union, Unsorted Union

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References

Armstrong, M. A. Basic Topology, rev. ed. New York: Springer-Verlag, 1997.Prasolov, V. V. Elements of Combinatorial and Differential Topology. Providence, RI: Amer. Math. Soc., p. xii, 2006.

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Disjoint Union

Cite this as:

Weisstein, Eric W. "Disjoint Union." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DisjointUnion.html

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