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:
“Willing sets you free: that is the true doctrine of will and freedom—thus Zarathustra instructs you.”
—Friedrich Nietzsche (1844–1900)
“Being dismantled before our eyes are not just individual programs that politicians cite as too expensive but the whole idea that society has a stake in the well-being of children down the block and the security of families on the other side of town. Whether or not kids eat well, are nurtured and have a roof over their heads is not just a consequence of how their parents behave. It is also a responsibility of society—but now apparently a diminishing one.”
—Richard B. Stolley (20th century)