Types of Sets — Definition, Examples & Symbols

TL;DR

A set is a well-defined collection of distinct objects, and its type is decided by how many elements it holds and how those elements relate to other sets. The main types of sets are finite, infinite, empty, singleton, equal, equivalent, subset, universal, and power set. This article defines each one with its symbol, draws the Venn-diagram picture, works through examples, and flags the comparisons students mix up most.

What Are the Main Types of Sets?

There are nine types worth naming. The first group is about size (how many elements), the second about relationships (how one set sits against another).

By size:

• Empty (null) set: has no elements. Written ∅ or \varnothing. Example: the set of real solutions to x² = −1 is ∅.

• Singleton (unit) set: has exactly one element. Example: {0}, or the set of even prime numbers {2}.

• Finite set: has a countable, definite number of elements. Example: {a, e, i, o, u} has five elements.

• Infinite set: has unending elements. Example: the natural numbers \mathbb{N} = {1, 2, 3, …}.

By relationship:

• Equal sets: same elements, regardless of order or repetition: {1, 2, 3} = {3, 1, 2}.

• Equivalent sets: same number of elements, not necessarily the same elements: {a, b, c} and {1, 2, 3} are equivalent because both have three.

• Subset: every element of one set is also in another: {1, 2} ⊆ {1, 2, 3}.

• Universal set: the set U containing every element under discussion; all other sets are subsets of it.

• Power set: the set of all subsets of a set, written P(A).

Two sets that share no elements at all, like {1, 2} and {5, 6}, are disjoint, and two that share some but not all are overlapping. Those two relationships sit naturally on top of the nine above.

Is the empty set finite?

Yes. The empty set ∅ has zero elements, and zero is a definite count, so ∅ is finite, and it is a subset of every set.

Examples of Types of Sets

Example 1

Classify A = {x : x is a month with 32 days}.
No month has 32 days. So A = ∅, an empty set.
Final answer: empty (null) set.

Example 2

A student is asked whether {2, 4, 6} and {2, 2, 4, 6, 6} are equal.
Take a closer look. A set holds only distinct objects, so repeats collapse: {2, 2, 4, 6, 6} is just {2, 4, 6}.
Final answer: the sets are equal, repetition does not create new elements.

Example 3

Are X = {vowels in "EQUAL"} and Y = {1, 2, 3} equivalent?
List X: the vowels in "EQUAL" are E, U, A, so X = {E, U, A}. Y has three elements.
Final answer: X and Y are equivalent but not equal.

Example 4

Is {1, 2} a subset of {1, 2, 3, 4}? Is it a proper subset?
Every element is present, so {1, 2} ⊆ {1, 2, 3, 4}.
Because {1, 2, 3, 4} has elements not in {1, 2}, it is also a proper subset: {1, 2} ⊂ {1, 2, 3, 4}.
Final answer: yes to both.

Example 5

Write the power set of B = {a, b} and state its size.
List every subset of B: ∅, {a}, {b}, {a, b}.
Count: 4 subsets.
Final answer: P(B) = {∅, {a}, {b}, {a, b}} with 4 elements.

Example 6

For the universal set U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, name the type of A relative to U, and find the complement of A.
Every element of A lies in U, so A is a subset of the universal set U.
The complement A’ holds everything in U that is not in A: A’ = {1, 3, 5}.
Final answer: A ⊆ U; A’ = {1, 3, 5}.

Why Naming the Type Comes Before Anything Else

• The empty set anchors logic. A claim with no examples resolves to ∅, and ∅ being a subset of everything is what lets proofs start from nothing and build up.
• Finite versus infinite decides your method. You can list a finite set; you must describe an infinite one.
• Equivalence underpins counting itself.

Cantor pushed exactly this idea to its edge and proved that the power set of any set is strictly larger than the set.

The Comparisons Students Mix Up Most

Mistake 1: Treating "equal" and "equivalent" as the same word

Don't do this: see two sets of the same size and call them equal.
The correct way: equal demands the same elements; equivalent demands only the same count.

Mistake 2: Thinking the empty set and the singleton ∅ are the same

Don't do this: write ∅ = {∅}.
The correct way: ∅ has zero elements; {∅} has one element. The cleanest tell: the empty set is the single thing you can drop into another set without adding a number to the count.

Mistake 3: Forgetting the universal set when taking a complement

Don't do this: the rusher computes A’ without writing down U first.
The correct way: a complement only means something relative to a fixed U.

Conclusion

• The types of sets split into size-based names (empty, singleton, finite, infinite) and relationship-based names (equal, equivalent, subset, universal, power set).
• A set holds only distinct elements, so repetition never changes a set's identity.
• Equal means same elements; equivalent means same count, the single most-confused pair.
• The empty set ∅ is finite and is a subset of every set; {∅} is a singleton, not the empty set.
• A complement only exists relative to a stated universal set.

Frequently Asked Questions

What are the main types of sets in mathematics?
The main types are empty (null), singleton, finite, infinite, equal, equivalent, subset, universal, and power set.

Is an empty set a finite set?
Yes. It has zero elements, and zero is a definite count.

What is the difference between equal and equivalent sets?
Equal sets have exactly the same elements; equivalent sets only have the same number of elements.

How many types of sets are there?
Most syllabi name nine core types.