Disjoint union: Difference between revisions
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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 \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, 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. |
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 \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, 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. |
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Latest revision as of 16:52, 25 March 2024
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 \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with ordinal sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, 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.