What is a subset in math?
A subset is a set whose elements all belong to another set. If every element of A is also in B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, the empty set is a subset of every set, and a set with n elements has 2n subsets.
Subset in a picture
Formal definition and notation
Set A is a subset of set B, written A ⊆ B, when every x ∈ A also satisfies x ∈ B. If in addition B has at least one element that is not in A, then A is a proper subset of B, written A ⊂ B. Two sets are equal exactly when each is a subset of the other.
| Symbol | Read it as | Meaning |
|---|---|---|
| A ⊆ B | A is a subset of B | every element of A is in B; A may equal B |
| A ⊂ B | A is a proper subset of B | A ⊆ B and A ≠ B |
| ∅ | the empty set | a subset of every set |
| 2n | two to the n | the number of subsets of a set with n elements |
Key facts
- To test A ⊆ B, check every element of A. One element of A missing from B is enough to say no.
- The empty set ∅ is a subset of every set, and every set is a subset of itself.
- A set with 3 elements has 23 = 8 subsets and 8 − 1 = 7 proper subsets.
- Use ∈ for an element and ⊆ for a set: 3 ∈ {1, 2, 3} but {3} ⊆ {1, 2, 3}.
- In probability, every event is a subset of the sample space: rolling an even number is {2, 4, 6} ⊆ {1, 2, 3, 4, 5, 6}.
Counting subsets
- n
- the number of elements in the original set
- 2n − 1
- the number of proper subsets (every subset except the set itself)
Each element is either in or out of a subset: 2 choices per element, so 2 × 2 × ⋯ × 2 = 2n subsets.
Subset examples, step by step
Example 1: Is it a subset?
Is {2, 5} a subset of {1, 2, 3, 4, 5}? Is {2, 6}?
- Check each element of {2, 5}: 2 is in the big set and 5 is in it. Yes.
- Check {2, 6}: 2 is in the big set, but 6 is not. No.
Answer: Yes, {2, 5} ⊆ {1, 2, 3, 4, 5}; no, {2, 6} is not a subset because 6 is missing.
Example 2: List every subset
List all subsets of {red, blue}. How many are proper subsets?
- Size 0: ∅.
- Size 1: {red} and {blue}.
- Size 2: {red, blue}, the set itself.
- Total = 22 = 4. Leave out the set itself to get the proper subsets: 3.
Answer: There are 4 subsets, and 3 of them are proper subsets.
Example 3: Pizza toppings
A pizza shop offers 5 toppings. A customer can choose any group of toppings, including none or all 5. How many different topping choices are possible?
- Each topping choice is a subset of the 5-topping set.
- Each topping is either on or off: 2 choices per topping.
- Total: 25 = 32, which counts the plain pizza (∅) and the everything pizza.
Answer: There are 32 possible topping choices.
Common mistakes
The empty set and the set itself are always subsets. Leaving them out is the most common counting error.
The subset symbol compares two sets. A single number is an element, so it takes ∈; wrap it in braces to make it a set.
Sets ignore order. Only membership matters when you test for a subset.
Not to be confused with
Every proper subset is a subset, but a set is a subset of itself without being a proper subset. {a, b} ⊆ {a, b} is true; {a, b} ⊂ {a, b} is false.
An element is a single member (4 ∈ {2, 4}); a subset is a whole set inside another ({4} ⊆ {2, 4}).
A set is any well-defined collection. Subset is a relationship between two sets, like taller than is a relationship between two people.
Quick check: 3 questions
How many subsets does {1, 2, 3, 4} have?
Answer: C. A 4-element set has 24 = 16 subsets. 15 is the number of proper subsets, 8 is 23 for a 3-element set, and 4 counts only the one-element subsets.Show answer and explanation
Which statement is true?
Answer: A. The empty set is a subset of every set. B uses ⊆ with an element, C fails because 7 is not in {5, 6}, and D fails because no set is a proper subset of itself.Show answer and explanation
A sample space is S = {1, 2, 3, 4, 5, 6}. Which set is the event roll a number greater than 4, as a subset of S?
Answer: B. Greater than 4 means 5 and 6, so the event is {5, 6} ⊆ S. A includes 4, C is the opposite event (the complement), and D misses 5.Show answer and explanation
Subset: frequently asked questions
Is the empty set a subset of every set?
Yes. To fail to be a subset, a set needs an element that is missing from the other set. The empty set has no elements, so it never fails. That is why ∅ ⊆ A for every set A, including ∅ itself, and why the empty set is always one of the 2n subsets you count.
What is the difference between a subset and a proper subset?
A subset may be the whole set; a proper subset must leave out at least one element. For B = {x, y}, the subsets are ∅, {x}, {y} and {x, y}, while the proper subsets are only the first three. The symbol ⊆ allows equality and ⊂ does not.
Why does a set with n elements have 2 to the n subsets?
Build a subset by going through the elements one at a time and deciding in or out. That is 2 choices for each of the n elements, and by the counting principle the choices multiply: 2 × 2 × ⋯ × 2 = 2n. For 3 elements there are 8 subsets.
Practice and tools
Sources
- OpenStax, Contemporary Mathematics, 1.2 Subsets
- Common Core State Standards, High School Statistics: Conditional Probability (S-CP.A.1)
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-CP.A.1
Cite this entry
Effortless Math Team. “Subset.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/subset/
Flashcard
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