Beyond subset containment, sets can relate to one another in several ways. Two sets may be equal, equivalent in size, disjoint, or overlapping. Understanding these relationships clarifies how collections compare and interact, and leads to the concept of a partition — a way of dividing a set into non-overlapping pieces.
Equal Sets
Two sets A and B are equal if they contain exactly the same elements:
A=B⟺∀x(x∈A⟺x∈B)
Order does not matter: {1,2,3}={3,1,2}. Repetition does not matter: {1,2,2,3}={1,2,3}. Only membership determines equality.
The standard method for proving two sets are equal is to show mutual subset containment:
A=B⟺(A⊆B)∧(B⊆A)
This reduces the problem to two subset proofs: show every element of A belongs to B, then show every element of B belongs to A.
Equivalent Sets
Two sets are equivalent if they have the same cardinality:
∣A∣=∣B∣
This means a bijection exists between A and B — a one-to-one correspondence pairing each element of A with exactly one element of B.
Equal sets are always equivalent: if A=B, then certainly ∣A∣=∣B∣. However, equivalent sets need not be equal.
The sets {a,b,c} and {1,2,3} are equivalent (both have cardinality 3) but not equal (they contain different elements). The sets N and Z are equivalent (both countably infinite) but not equal (Z contains negative integers).
Disjoint Sets
Two sets are disjoint if they have no elements in common:
A∩B=∅
The sets {1,2,3} and {4,5,6} are disjoint. The set of even integers and the set of odd integers are disjoint.
For a collection of more than two sets, the sets are pairwise disjoint if every pair is disjoint:
Ai∩Aj=∅for all i=j
Pairwise disjoint sets have no overlap whatsoever — no element belongs to more than one set in the collection.
In probability, disjoint events are called mutually exclusive: if one occurs, the others cannot.
Overlapping Sets
Two sets overlap if they share at least one element but neither is a subset of the other:
A∩B=∅andA⊆BandB⊆A
This means each set contains elements the other lacks, yet they also have common elements.
The sets {1,2,3} and {2,3,4} overlap: they share 2 and 3, but 1 belongs only to the first and 4 belongs only to the second.
In a Venn diagram, overlapping sets appear as circles that intersect but neither contains the other. The lens-shaped intersection region is non-empty, and both the A∖B and B∖A regions are also non-empty.
Partitions
A partition of a set S is a collection of non-empty, pairwise disjoint subsets whose union equals S:
{A1,A2,…,An} partitions S⟺⎩⎨⎧Ai=∅Ai∩Aj=∅A1∪A2∪⋯∪An=Sfor all ifor i=j
Every element of S belongs to exactly one subset in the partition.
Examples:
{{1,3},{2,4},{5}} partitions {1,2,3,4,5}
Z
n partition Z into n subsets
Partitions arise naturally in classification problems and connect to equivalence relations. The law of total probability relies on partitions: if events B1,B2,…,Bn partition the sample space, probabilities can be computed by summing over the partition.