Math
Subset Calculator
Type two sets A and B and the subset calculator tests A ⊆ B (every element of A is in B), tells you whether the subset is proper (B has something extra), checks the reverse direction, and reports equality.
Any element of A missing from B is listed so you can see exactly why a relation fails.
Separate elements with commas or spaces. Braces are optional.
The set that may contain A.
Try an example
Result
Relation
—
- A ⊆ B?
- —
- A ⊂ B (proper)?
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- B ⊆ A?
- —
- A = B?
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- Elements of A not in B
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- Common elements A ∩ B
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More details (2 more)
- Sizes
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- Number of subsets of A
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the subset calculator solves
A is a subset of B, written A ⊆ B, when nothing in A is missing from B. If in addition B has at least one element that A lacks, A is a proper subset, A ⊂ B. If each set is a subset of the other, they are equal. The empty set is a subset of every set, and every set is a subset of itself.
| A | B | Relation |
|---|---|---|
| {1, 2} | {1, 2, 3, 4} | A ⊂ B (proper subset) |
| {1, 2, 3} | {3, 2, 1} | A = B (order does not matter) |
| {1, 5} | {1, 2, 3} | A ⊄ B (5 is missing) |
| ∅ | {a} | ∅ ⊂ {a} |
How to use it
- Type set A and set B with commas or spaces between elements.
- Read the relation at the top: proper subset, equal, reversed, or not a subset.
- Check the missing-elements line to see which elements break the relation.
How to read the answer
A ⊆ B means A fits inside B; A ⊂ B adds that B is strictly bigger. 'B ⊆ A' tells you whether the relation holds the other way round. When both hold the sets are equal. The common-elements line is the intersection A ∩ B.
Common mistakes and edge cases
- Thinking order or repetition matters: {1, 2, 2} and {2, 1} are the same set.
- Saying a set is a proper subset of itself; A ⊂ A is false, though A ⊆ A is true.
- Forgetting that ∅ is a subset of every set.
- Confusing ∈ (is an element of) with ⊆ (is a subset of): 1 ∈ {1, 2} but {1} ⊆ {1, 2}.
Worked examples
Proper subset
{1, 2} ⊂ {1, 2, 3, 4}
Relation
A ⊂ B (A is a proper subset of B)
Equal sets in different order
{1, 2, 3} = {3, 2, 1}
Relation
A = B (equal sets; each is a subset of the other, neither is a proper subset)
Not a subset
{1, 5} ⊄ {1, 2, 3} because 5 is missing
Relation
A ⊄ B (A is not a subset of B)
Reverse relation
{1, 2, 3} is not inside {1, 2}, but {1, 2} ⊂ {1, 2, 3}
Relation
B ⊂ A (B is a proper subset of A; A is not a subset of B)
Empty set
∅ is a proper subset of {a}
Relation
A ⊂ B (A is a proper subset of B)
Letters
{x, y} ⊂ {w, x, y, z}
Relation
A ⊂ B (A is a proper subset of B)
Frequently asked questions
What is the difference between ⊆ and ⊂?+
A ⊆ B allows A = B; A ⊂ B (proper subset) requires B to have at least one extra element. Some books write ⊂ for both, so check the convention in use.
Is the empty set a subset of every set?+
Yes. There is no element of ∅ that could fail to be in B, so the condition holds automatically (vacuously).
Is a set a subset of itself?+
Yes, A ⊆ A always. It is not a proper subset of itself.
Does the order of elements matter?+
No. {1, 2, 3} and {3, 1, 2} are the same set. Repeated elements are also ignored.
How many subsets does a set have?+
2^n for n elements, including ∅ and the set itself. The calculator shows this count for A.
What is a superset?+
If A ⊆ B, then B is a superset of A, written B ⊇ A. The 'B ⊆ A' line tells you whether A is a superset of B.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026