Functions, Sets and Families
Surjective functions and families are formally equivalent, as any function f with domain I induces a family (f(i))i∈I. In practice, however, a family is viewed as a collection, not as a function: being an element of a family is equivalent with being in the range of the corresponding function. A family contains any element exactly once, if and only if the corresponding function is injective.
Like a set, a family is a container and any set X gives rise to a family (xx)x∈X. Thus any set naturally becomes a family. For any family (Ai)i∈I there is the set of all elements {Ai | i∈I}, but this does not carry any information on multiple containment or the structure given by I. Hence, by using a set instead of the family, some information might be lost.
Read more about this topic: Indexed Family
Famous quotes containing the words sets and/or families:
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—Samuel Johnson (1709–1784)
“Awareness has changed so that every act for children, every piece of legislation recognizes that children are part of families and that it is within families that children grow and thrive—or don’t.”
—Bernice Weissbourd (20th century)