Math 2.0

Mathematical logic and foundations Definition

Partition of a set

Builds on SubsetDisjoint setsUnion of a family of sets

Definition

Let AA be a set. A partition of AA is a set PP of subsets of AA, called blocks, with these three properties:

  1. each block is nonempty;
  2. the blocks are pairwise disjoint;
  3. the union of the blocks is AA.

Equivalently, each element of AA is in exactly one block.

Remarks

The set {{1,3},{2}}\{\{1, 3\}, \{2\}\} is a partition of {1,2,3}\{1, 2, 3\}. The even and the odd integers form a partition of ℤ\mathbb{Z} with two blocks. The equivalence classes form a partition, and each partition comes from exactly one equivalence relation. See the correspondence between equivalence relations and partitions.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionPartition of a set, definition, no evidence levelPartition of a setdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Intersection definition
  11. Union definition
  12. Existential quantifier definition
  13. Universal quantifier definition
  14. Disjoint sets definition
  15. Subset definition
  16. Union of a family of sets definition
  17. Partition of a set definition

Where this leads

57 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Partition_of_a_set, lead section
About this page
Tag09CCE86A6D
URIm2://09CCE86A6D
Kinddefinition
WritingW1 stated
MSC03E20
DefinesPartition of a set
UsesSubset, Disjoint sets, Union of a family of sets

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by