Partition (abstract algebra)
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Definition
Let [ilmath]I[/ilmath] be an arbitrary indexing set, to each element [ilmath]i\in I[/ilmath] we assign a set [ilmath]A_i[/ilmath] which is non-empty. If:
- [ilmath]A_i\cap A_j=\emptyset[/ilmath] for [ilmath]i\ne j[/ilmath] (the [ilmath]A_i[/ilmath] are mutually disjoint)
- [ilmath]B=\bigcup_{i\in I}A_i[/ilmath]
Then the family [ilmath]\{A_i\}_{i\in I} [/ilmath] is called[1] the partition of the set [ilmath]B[/ilmath] into classes [ilmath]A_i[/ilmath] for [ilmath]i\in I[/ilmath]
Equality
- Two partitions are identical (and can be swapped around as needed) if they have the same indexing family and the same set assigned to each element of the indexing family[2]
Subpartition
- We say the partition [ilmath]\{C_j\}_{j\in J} [/ilmath] is finer than [ilmath]\{A_i\}_{i\in I} [/ilmath] (or a subpartition of [ilmath]\{A_i\}_{i\in I} [/ilmath]) if we have:
- [math]\forall j\in J\exists i\in I[C_j\subseteq A_i][/math] (Or in Krzysztof Maurin's notation [math]\bigwedge_{j\in J}\bigvee_{i\in I}C_j\subseteq A_i[/math])[Note 1]
See also
Notes
- ↑ The book uses a strict [ilmath]\subset[/ilmath] however take [1,2,3], [4,5] as a partition of 1-5, then [1],[2,3],[4,5] is a sub-partition, but [4,5][ilmath]\not\subset[/ilmath][4,5] - however [4,5][ilmath]\subseteq[/ilmath][4,5]
References
- ↑ Analysis - Part 1: Elements - Krzystof Maurin
- ↑ Alec's own work - equality is a difficult definition as the partition sets may be associated with different members in the indexing set, this could be important. Example, [1,2,3] and [4,5] partition 1-5, we could associate [ilmath]i\in I[/ilmath] with [1,2,3] and also [ilmath]j\in J[/ilmath] with [1,2,3] but there's no requirement for [ilmath]i=j[/ilmath] -it would be naive to consider these equal if [ilmath]i\ne j[/ilmath]
