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Subset

Check if A⊆B or A⊂B. List all subsets, count, intersection, union, complement. Set theory.

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Why: Understanding subset helps you make better, data-driven decisions.

How: Enter Set A (comma or space separated), Set B (comma or space separated) to calculate results.

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📐 Examples — Click to Load

Check:
subset_check.sh
CALCULATED
Set A
{1, 2}
Set B
{1, 2, 3}
A ⊆ B
Yes
A ⊂ B
Yes
A ∩ B
{1, 2}
A ∪ B
{1, 2, 3}
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Set Sizes

Overlap

Set A = {1, 2}, Set B = {1, 2, 3}
A ⊆ B (every element of A in B)? Yes
A ⊂ B (proper subset, A ≠ B)? Yes
A ⊇ B (B is subset of A)? No
A ∩ B = {1, 2}
A ∪ B = {1, 2, 3}
B \ A (complement of A in B) = {3}

For educational and informational purposes only. Verify with a qualified professional.

⊆ vs ⊂ Notation

A ⊆ B means A is a subset of B (every element of A is in B; A may equal B). A ⊂ B means A is a proper subset (A ⊆ B and A ≠ B). Some texts use ⊂ for both; we distinguish them here.

∅ is Subset of All Sets

The empty set ∅ has no elements, so "every element of ∅ is in B" is vacuously true. Thus ∅ ⊆ A for any set A.

📊 Venn Diagram Interpretation

A ⊆ B means the circle for A lies entirely inside the circle for B. A ∩ B is the overlap; A ∪ B is the combined region.

🔢 Cardinality Relationships

If A ⊆ B then |A| ≤ |B|. For the power set of a set with n elements, there are 2^n subsets. |A ∪ B| = |A| + |B| - |A ∩ B|.

📝 Worked Examples

{1,2} ⊆ {1,2,3} — Every element of {1,2} is in {1,2,3}. Yes. And {1,2} ⊂ {1,2,3} (proper).
∅ ⊆ {1,2,3} — Empty set has no elements to violate. Vacuously true. ∅ ⊆ any set.
Power set of {1,2} — 2² = 4 subsets: ∅, {1}, {2}, {1,2}.

📌 Summary

A ⊆ B means every element of A is in B. A ⊂ B means proper subset (A ≠ B). ∅ ⊆ A always. Power set has 2^n elements. |A∪B| = |A| + |B| - |A∩B|.

❓ FAQ

What is a subset?

A ⊆ B means every element of A is also in B. {1,2} ⊆ {1,2,3}.

What is a proper subset?

A ⊂ B means A ⊆ B and A ≠ B. So {1,2} ⊂ {1,2,3}, but {1,2} is not a proper subset of {1,2}.

Is the empty set a subset?

Yes. ∅ ⊆ A for any set A. The empty set is a subset of every set.

How many subsets does a set have?

A set with n elements has 2^n subsets (including ∅ and itself). This is the power set.

What is A ∩ B?

Intersection: elements in both A and B. {1,2} ∩ {2,3} = {2}.

What is A ∪ B?

Union: elements in A or B (or both). {1,2} ∪ {2,3} = {1,2,3}.

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