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partition

2024-01-01

let be a non-empty set and (a family of subsets of ), if the following conditions are met:

  1. for every (meaning every in isnt empty),
  2. for every where (every 2 sets of are different),
  3. ( is equal to the union of all the sets of ),

then is a partition of .

consider and , let , we check if the set matches the conditions

  1. for every , this checks out because
  2. for every where , this checks out because none of have any common elements
  3. , this checks out because

therefore is a partition of

if is an equivalence relation over then the quotient set is a partition of .

if is a partition of then the relation that is defined as: is an equivalence relation in and

a partition of a set is a collection of nonempty disjoint subsets of whose union is .