Disjoint union

The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). The cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar.