Indexed Family

First, a mapping ''f'' from a set ''J'' to a set ''X'' is alternatively conceptualized as a ''family of elements'' of ''X'' indexed by ''J''. In this usage, ''J'' is called the '' Index Set '' of the family ''f'' : ''J'' → ''X'', the functional image ''f''(''j'') for ''j'' ∈ ''J'' is denoted ''x''_''j'', and the mapping ''f'' is denoted {''x''_''j''}_{''j''∈''J''} or simply {''x''_''j''}. Next, if the set ''X'' is the Power Set of a set ''U'', then the family {''x''_''j''}_{''j''∈''J''} is called a family of sets indexed by ''J'' , or simply a '''family of sets'''. NOTATION A family $f:J ightarrow A$ can be denoted by $(A_j)_{j\in J}$ where $J$ is the index set and $j ightarrow A_j$ is the mapping. So $A_j$ is the element belonging to the key $j$ , also called the $j^{th}$ element of the family.
	Note That <math>\{A Jj\in J\}</math> Is A	"http://wwwinformationdelightinfo/information/entry/set" class="copylinks">Set
	Like A	"http://wwwinformationdelightinfo/information/entry/set" class="copylinks">Set , a family is a container and any set ''X'' gives rise to a family (''x'')<sub>''x''&isin''X''</sub> Thus any set naturally becomes a family For any family (''A''<sub>''i''</sub>)<sub>''i''&isin''I''</sub> there is the set of all elements {''A''<sub>''i''</sub> ''i''&isin''I''}, but this does not carry any information on multiple containment or the structure of ''I'' Hence, by using a set instead of the family, some information might be lost

	CATEGORIES ABOUT INDEXED FAMILY

	mathematical notation

	basic concepts in set theory

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