Page Title

Key Terms

Equal Sets

Summary: How Sets Relate

Set Relationships FAQ

How Sets Compare and Connect

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.

Key Terms

Equal Sets— sets containing exactly the same elements

Equivalent Sets— sets with the same cardinality

Disjoint Sets— sets with no common elements

Overlapping Sets— sets sharing at least one element

Partition— non-overlapping subsets that cover the whole set

Subset— equality is proved via mutual subset containment

See All Set Theory Definitions →

Equal Sets

Two sets

A

and

B

are equal if they contain exactly the same elements:

A = B \iff \forall x\,(x \in A \iff x \in B)

The two logic symbols in this criterion read naturally:

\forall

means 'for all' and

\iff

means 'if and only if'.

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:

Set Equality Criterion

A = B \iff (A \subseteq B) \land (B \subseteq A)

Learn more about this formula: Set Equality Criterion →

Browse all set theory formulas →

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

\mathbb{N}

and

\mathbb{Z}

are equivalent (both countably infinite) but not equal (

\mathbb{Z}

contains negative integers).

Disjoint Sets

Two sets are disjoint if they have no elements in common:

A \cap B = \emptyset

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:

A_i \cap A_j = \emptyset \quad \text{for all } i \neq 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 \cap B \neq \emptyset \quad \text{and} \quad A \not\subseteq B \quad \text{and} \quad B \not\subseteq 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 \setminus B

and

B \setminus 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

\{A_1, A_2, \ldots, A_n\} \text{ partitions } S \iff \begin{cases} A_i \neq \emptyset & \text{for all } i \\ A_i \cap A_j = \emptyset & \text{for } i \neq j \\ A_1 \cup A_2 \cup \cdots \cup A_n = S \end{cases}

Every element of

S

belongs to exactly one subset in the partition.

Examples:

$\{\{1, 3\}, \{2, 4\}, \{5\}\}$ partitions $\{1, 2, 3, 4, 5\}$

$\mathbb{Z}$

$n$ partition $\mathbb{Z}$ into $n$ subsets

Partitions arise naturally in classification problems and connect to equivalence relations. The law of total probability relies on partitions: if events

B_1, B_2, \ldots, B_n

partition the sample space, probabilities can be computed by summing over the partition.

Summary: How Sets Relate

The five relationships introduced above describe the main ways sets can be related to one another — from identity, through size equivalence, through varying degrees of overlap, to the multi-set case of partitioning. The capstone table below collects them all with the defining condition, a worked example, and the key fact that distinguishes each relationship from the others.

Relationship	Defining condition	Example	Key distinguishing fact
Equal sets	same elements: ∀x (x ∈ A ⇔ x ∈ B)	{1, 2, 3} = {3, 1, 2}	proved by mutual subset containment (A ⊆ B and B ⊆ A)
Equivalent sets	same cardinality: \|A\| = \|B\|, written A ~ B	{a, b, c} ~ {1, 2, 3}; ℕ ~ ℤ	equal ⇒ equivalent, but not the other way around
Disjoint sets	no common elements: A ∩ B = ∅	{1, 2, 3} and {4, 5, 6}; evens and odds	called "mutually exclusive" in probability
Overlapping sets	A ∩ B ≠ ∅, A ⊄ B, B ⊄ A	{1, 2, 3} and {2, 3, 4}	both A \ B and B \ A are non-empty (lens-shaped Venn intersection)
Partition (of S)	non-empty, pairwise disjoint subsets whose union = S	{evens, odds} of ℤ; residue classes mod n	every element of S belongs to exactly one piece

Set Relationships FAQ

What is the difference between equal and equivalent sets?

Equal sets contain exactly the same elements. Equivalent sets merely have the same size, so their elements can differ entirely. {a, b, c} and {1, 2, 3} are equivalent but not equal. Every pair of equal sets is equivalent, but the reverse fails. Equivalence is settled by cardinality, equality by membership.Read more →

What are disjoint sets?

Two sets are disjoint when they share no elements, so their intersection is empty. The even and odd integers are disjoint, as are {1, 2, 3} and {4, 5, 6}. In probability the same condition is called mutually exclusive: if one event happens, the other cannot. Disjointness is a property of the pair, not of either set alone.Read more →

What does pairwise disjoint mean?

A collection of three or more sets is pairwise disjoint when every possible pair of them is disjoint. Checking only that all the sets share no common element is not enough: three sets can have empty triple intersection while two of them still overlap. Pairwise disjoint is the stronger condition and the one partitions require.Read more →

How do you prove two sets are equal?

Show containment in both directions. Prove that every element of A belongs to B, then prove that every element of B belongs to A. Together these give A ⊆ B and B ⊆ A, which forces A = B. This double-inclusion method turns one equality claim into two subset proofs and is the standard technique.Read more →

What is a partition of a set?

A partition splits a set into non-empty pieces that are pairwise disjoint and together cover everything, so each element lands in exactly one piece. The even and odd integers partition the integers; residue classes mod n partition them into n pieces. The law of total probability is built directly on this idea.Read more →