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Fundamentals of Business Mathematics and Statistics · Set Theory, including Venn Diagram

Types of Sets with Examples and Subset Formulas

Updated 10 October 2026 · Fact-checked

Sets are classified by size and by how they relate to each other: empty, singleton, finite, infinite, equal, equivalent and universal sets, plus subsets and power sets. A set with n elements has 2ⁿ subsets, 2ⁿ − 1 proper subsets, and a power set of 2ⁿ elements.

Understand Types of Sets

A set is a well-defined collection of distinct objects. Each object is an element. Order does not matter and repeats are ignored, so {1, 2, 2, 3} is the same set as {3, 1, 2}.

First, classify by size. The empty set (∅ or { }) has no elements. A singleton set has exactly one element, like {5}. A finite set has a countable number of elements, like the days in a week. An infinite set never ends, like the natural numbers. Note that {0} is a singleton, not empty, because it contains the element 0. Also {∅} is a singleton, because it contains one element, ∅.

Next, compare two sets. Equal sets have exactly the same elements. Equivalent sets have the same number of elements, but the elements may differ. So {1, 2, 3} and {a, b, c} are equivalent but not equal. Every pair of equal sets is equivalent, but equivalent sets need not be equal.

Then comes the subset. A is a subset of B (A ⊆ B) if every element of A is also in B. A proper subset (A ⊂ B) is a subset that is not equal to B, so B has at least one element A lacks. The empty set is a subset of every set. Every set is a subset of itself.

The universal set (U) is the big set that contains all the elements under discussion. The power set P(A) is the set of all subsets of A, including ∅ and A itself. Its elements are sets, not numbers.

Key formulas to remember

Number of subsets
Number of subsets of A = 2ⁿ
n = number of elements in A. Includes ∅ and A itself.
Number of proper subsets
2ⁿ − 1
Excludes only the set A itself. Empty set is counted as a proper subset when A is non-empty.
Number of non-empty proper subsets
2ⁿ − 2
Excludes both ∅ and A.
Number of elements in power set
n[P(A)] = 2ⁿ
Same as the number of subsets.
Subsets of a given size
Number of subsets with r elements = nCr = n! ÷ (r! × (n − r)!)
Use when the question asks for subsets with exactly r elements.
Equal and equivalent sets
A = B if same elements; A ~ B if n(A) = n(B)
Equal implies equivalent, not the other way round.
Subset of an empty set
∅ has exactly 1 subset (itself) and 0 proper subsets
Here n = 0, so 2⁰ = 1 and 2⁰ − 1 = 0.

How to solve Types of Sets questions

Use this routine for any question on types of sets, subsets or power sets.

  1. 1Write the set out and remove repeated elements. Do this before counting anything.
  2. 2Count the distinct elements to get n. Remember {0} and {∅} each have 1 element.
  3. 3Identify what the question asks: type of set, equality or equivalence, subset or proper subset, or number of subsets.
  4. 4For counting subsets, apply 2ⁿ. For proper subsets, subtract 1. For non-empty proper subsets, subtract 2.
  5. 5For subsets of a fixed size, use nCr.
  6. 6For true/false or symbol questions, check element by element: ∈ is for an element in a set, ⊆ is for a set inside a set.
  7. 7Check edge cases: empty set, the set itself, and sets whose elements are themselves sets.

Quickest way: Count n, then use powers of 2

When to use it: Any MCQ asking for number of subsets, proper subsets or size of a power set.

  1. Count distinct elements n. Ignore repeats.
  2. Recall powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256.
  3. Subsets = 2ⁿ. Proper subsets = 2ⁿ − 1. Non-empty proper = 2ⁿ − 2.
  4. If the question gives the subset count, work backwards: find n such that 2ⁿ equals it.
  5. Use elimination: the answer for subsets is always a power of 2, so discard options that are not.

Common mistakes in Types of Sets

  • Counting repeated elements when finding n

    Students read the list as written and do not clean it up.

    Fix: Rewrite the set with distinct elements first. {1, 2, 2, 3} has n = 3.

  • Treating {0} or {∅} as the empty set

    Zero and the symbol ∅ look like 'nothing'.

    Fix: Both contain one element, so each is a singleton with n = 1.

  • Confusing equal and equivalent sets

    Both words suggest 'the same'.

    Fix: Equal means same elements. Equivalent means same count only. {1, 2} and {5, 6} are equivalent, not equal.

  • Giving 2ⁿ as the number of proper subsets

    Students forget that the set itself is not a proper subset.

    Fix: Proper subsets = 2ⁿ − 1. Always subtract the set itself.

  • Mixing up ∈ and ⊆

    Both mean 'belongs to' in everyday thinking.

    Fix: Use ∈ for an element, ⊆ for a set. For A = {1, 2}: 1 ∈ A but {1} ⊆ A.

  • Leaving out ∅ or the set itself when listing a power set

    Students list only the 'visible' subsets.

    Fix: Always include ∅ and A. Check that your list has 2ⁿ members.

Worked examples

Example 1

If A = {2, 4, 4, 6, 8, 8}, find the number of subsets and the number of proper subsets of A.

Show the solution
  1. Remove repeats: A = {2, 4, 6, 8}.
  2. So n = 4.
  3. Number of subsets = 2⁴ = 16.
  4. Number of proper subsets = 16 − 1 = 15.

Answer: 16 subsets and 15 proper subsets.

Example 2

A set has 127 proper subsets. How many elements does it have, and how many elements are in its power set?

Show the solution
  1. Proper subsets = 2ⁿ − 1 = 127.
  2. So 2ⁿ = 128.
  3. Since 2⁷ = 128, n = 7.
  4. Power set has 2ⁿ = 128 elements.

Answer: The set has 7 elements and its power set has 128 elements.

Exam tips

  • Memorise powers of 2 up to 2¹⁰ = 1024. Most subset questions are solved by recall.
  • Read the wording carefully: subsets, proper subsets and non-empty proper subsets give three different answers.
  • Watch for traps like {0}, {∅} and sets with repeated elements before counting.
  • For symbol questions, test each statement by asking whether the left side is an element or a set.
  • With no negative marking, always attempt every question. Eliminate non-powers of 2 for subset counts.

Practice questions from Set Theory, including Venn Diagram

Types of Sets: frequently asked questions

What is the difference between a subset and a proper subset?

A subset can be equal to the original set, so A ⊆ A is true. A proper subset must be smaller, so it leaves out at least one element of the original set. A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.

What is the difference between equal sets and equivalent sets?

Equal sets contain exactly the same elements. Equivalent sets contain the same number of elements, which may be different. For example, {1, 2} and {a, b} are equivalent but not equal.

How do I find the number of subsets of a set?

Count the distinct elements n and compute 2ⁿ. For {a, b, c}, n = 3, so there are 2³ = 8 subsets. This count includes the empty set and the set itself.

Is the empty set a subset of every set?

Yes. The empty set has no element that could be missing from another set, so it is a subset of every set. It is also a proper subset of every non-empty set.