Category Theory Point of View
In category theory the disjoint union is defined as a coproduct in the category of sets.
As such, the discrete union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead.
This categorical aspect of the discrete union explains why is frequently used, instead of, to denote it.
Read more about this topic: Disjoint Union
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