When are two sets equal?

Two sets are equal if they contain exactly the same elements. Order and repetition don't matter — only membership. Prove equality by showing A ⊆ B and B ⊆ A.

What is the difference between equal and equivalent sets?

Equal sets have identical elements. Equivalent sets have the same cardinality (size) but may contain different elements. Equal implies equivalent, but not vice versa.

What are disjoint sets?

Two sets are disjoint if they share no elements — their intersection is empty: A ∩ B = ∅. In probability, disjoint events are called mutually exclusive.

What does pairwise disjoint mean?

A collection of sets is pairwise disjoint if every pair has empty intersection: Aᵢ ∩ Aⱼ = ∅ for all i ≠ j. No element belongs to more than one set.

What are overlapping sets?

Sets overlap when they share at least one element but neither is a subset of the other. Each contains elements the other lacks, yet their intersection is non-empty.

What is a partition of a set?

A partition divides a set S into non-empty, pairwise disjoint subsets whose union equals S. Every element of S belongs to exactly one subset in the partition.

How do you prove two sets are equal?

Show mutual subset containment: prove every element of A belongs to B, then prove every element of B belongs to A. This establishes A = B.

Can equivalent sets have different elements?

Yes. Equivalent sets only need the same cardinality. For example, {a, b, c} and {1, 2, 3} are equivalent (both size 3) but not equal (different elements).

How are partitions used in probability?

If events B₁, B₂, ..., Bₙ partition the sample space, the law of total probability lets you compute P(A) by summing P(A|Bᵢ)P(Bᵢ) over the partition.

What is an example of a partition?

The even and odd integers partition ℤ. Residue classes modulo n partition ℤ into n subsets. Each integer belongs to exactly one class.

Page Title

Summary: How Sets Relate

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)

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

\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\|	{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