Subsets - When One Set Lives Inside Another

A subset is a set whose every element also belongs to another set. If every member of set A is also a member of set B, then A is a subset of B. Understanding subsets is essential for describing relationships between groups and forms the backbone of set operations.

Notation

A ⊆ B means “A is a subset of B” (every element of A is in B).
A ⊂ B means “A is a proper subset of B” (A ⊆ B and A ≠ B, i.e. B has at least one element not in A).
A ⊈ B means “A is not a subset of B” (at least one element of A is not in B).

Key Facts About Subsets

Every set is a subset of itself: A ⊆ A.
The empty set is a subset of every set: ∅ ⊆ A for any set A.
If A ⊆ B and B ⊆ A, then A = B.
A set with n elements has exactly 2ⁿ subsets (including ∅ and the set itself).

Listing All Subsets

To list every subset of a set, work systematically: start with ∅, then all single-element subsets, then all two-element subsets, and so on up to the full set.

Worked Examples

A = {1, 2, 3}. List all subsets of A and count them.

Subsets with 0 elements: ∅
Subsets with 1 element: {1}, {2}, {3}
Subsets with 2 elements: {1,2}, {1,3}, {2,3}
Subsets with 3 elements: {1,2,3}
Total: 8 subsets = 2³ ✓

B = {2, 4, 6, 8, 10}. Is {2, 6} ⊆ B? Is {2, 5} ⊆ B? Is B ⊂ B?

{2, 6} ⊆ B: True – both 2 and 6 are in B.
{2, 5} ⊆ B: False – 5 is not in B.
B ⊂ B: False – B ⊆ B is true, but B ⊂ B (proper subset) requires B ≠ B, which is never true.

How many subsets does the set {a, b, c, d} have? How many are proper subsets?

n = 4 elements. Total subsets = 2⁴ = 16.
Proper subsets (excluding the set itself) = 16 − 1 = 15.

P = {even numbers}, Q = {multiples of 4}. Is Q ⊆ P? Is P ⊆ Q?

Every multiple of 4 (4, 8, 12, …) is even, so Q ⊆ P (in fact Q ⊂ P).
P ⊆ Q: False – 2 is even but not a multiple of 4.

Power Set

The power set of A, written P(A), is the set of all subsets of A.
If |A| = n, then |P(A)| = 2ⁿ.

Example: A = {x, y}. P(A) = {∅, {x}, {y}, {x,y}}. |P(A)| = 4 = 2².

Key Takeaways

A ⊆ B: every element of A is also in B.
A ⊂ B (proper subset): A ⊆ B and A ≠ B.
∅ is a subset of every set; every set is a subset of itself.
A set with n elements has 2ⁿ subsets total and 2ⁿ − 1 proper subsets.

Practice Questions

List all subsets of {p, q}.
How many subsets does {1, 2, 3, 4, 5} have?
True or false: (a) {3, 5} ⊆ {1, 2, 3, 4, 5}; (b) ∅ ⊂ {7}; (c) {1, 2} ⊂ {1, 2}.
A = {multiples of 6 up to 30}, B = {multiples of 2 up to 30}. Is A ⊆ B? Is B ⊆ A?
Write the power set of {a, b, c} and verify the count using 2ⁿ.

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