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:
“There be some sports are painful, and their labor
Delight in them sets off. Some kinds of baseness
Are nobly undergone, and most poor matters
Point to rich ends.”
—William Shakespeare (1564–1616)
“Whatever else American thinkers do, they psychologize, often brilliantly. The trouble is that psychology only takes us so far. The new interest in families has its merits, but it will have done us all a disservice if it turns us away from public issues to private matters. A vision of things that has no room for the inner life is bankrupt, but a psychology without social analysis or politics is both powerless and very lonely.”
—Joseph Featherstone (20th century)