Family
The formal term used for a collection of objects. It is denoted ![<span style=]()
{a_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline1.svg" style="height:24px; width:46px" /> (but other kinds of brackets can be used as well), where 
is a nonempty set called the index set, and 
is called the term of index 
of the family.
A family with index set 
is called a sequence.
The union and the intersection of a family of sets ![<span style=]()
{A_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline6.svg" style="height:24px; width:49px" /> are denoted
 |
(1)
|
respectively.
If all terms 
belong to an additive monoid, one can consider the sum
 |
(2)
|
provided the number of nonzero terms is finite, i.e., the so-called support of the family
![<span style=]() {i in I|a_i!=0} " src="https://mathworld.wolfram.com/images/equations/Family/NumberedEquation3.svg" style="height:22px; width:94px" /> |
(3)
|
is a finite set. A similar argument applies to multiplicative monoids, and to the product
 |
(4)
|
up to replacement of the zero element with the identity element 1.
According to its formal definition (Bourbaki 1970), if the terms 
belong to the set 
, the family ![<span style=]()
{a_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/Family/Inline10.svg" style="height:24px; width:46px" /> is a map 
, where 
for all 
.
Every set 
gives rise to a family
 |
(5)
|
from which the original set can be recovered as the range of 
. Accordingly, every family 
also gives rise to a set
![X=<span style=]() {a_i|i in I}, " src="https://mathworld.wolfram.com/images/equations/Family/NumberedEquation6.svg" style="height:22px; width:106px" /> |
(6)
|
from which, however, the original family in general cannot be recovered.
REFERENCES
Bourbaki, N. Eléments de Mathématiques. Théorie des Ensembles. Paris, France: Hermann, p. ER11, 1970.
