In these lessons, we will learn the concept of subsets and proper subsets and the formula for the number of subsets in a finite set.

**Related Pages**

More Lessons On Sets

Universal Set

Venn Diagrams

If every member of set A is also a member of set B, then A is a **subset** of B, we write A ⊆ B.
We can say A is contained in B.

We can also say B ⊇ A, B is a superset of A, B includes A, or B contains A.

If A is not a subset of B, we write A ⊈ B.

If A is a subset of B (A ⊆ B), but A is not equal to B, then we say A is a **proper subset** of B,
written as A ⊂ B or A ⊊ B.

The following diagram shows an example of subset. Scroll down the page for more examples and solutions on subsets.

**Example:**

A = {1, 3, 5}, B = {1, 2, 3, 4, 5}, C = {1, 2, 3, 4, 5}

A is a subset of B, A ⊆ B. because every element in A is also in B

A is also proper subset of B, A ⊂ B. because every element in A is also in B and A ≠ B

C is subset of B, C ⊆ B. but is not a proper subset of B because C = B

**Example:**

X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.

X is not a subset of Y, X ⊈ Y, because the element 1 is in X but not in Y.

Note:

• The empty set denoted by ∅ or {} is a subset of any set.

∅ ⊆ S

• The empty set is a proper subset of all sets except ∅

∅ ⊂ S ≠ ∅

**Example:**

List all the subsets and proper subsets of the set Q = {x, y, z}

**Solution:**

The subsets of Q are { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z} and {x, y, z}

The proper subsets of Q are { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}

The number of subsets for a finite set A is given by the formula:

If set A has n elements, it has 2^{n} subsets.

If set A has n elements, it has 2^{n} - 1 proper sets.

**Example:**

Q = {x, y, z}. How many subsets and proper subsets will Q have?

**Solution:**

Q has 3 elements

Number of subsets = 2^{3 }= 8

Number of proper subsets = 7

**Example:**

Draw a Venn diagram to represent the relationship between the sets. A = {1, 3, 5} and B = {1, 2, 3, 4, 5}

**Solution:**

Since A is a subset of B:

Step 1: Draw circle A within the circle B

Step 2: Write down the elements in circle A.

Step 3: Write down the remaining elements in circle *B*

This video defines and give the notation or symbols used for subsets and proper subsets and shows how to determine the number of possible subsets for a given set.

**Example:**

Given the set C = {1, 2, 3}, what are the subsets and proper subsets?

**Example:**

Indicate whether true or false:

{} ⊆ {2, 3}

{} ∈ {2, 3}

{} ∈ {{}, 2, 3}

{5, 6, 7} ⊆ {5, 6, 7, 8}

{5, 6, 7, 8} ⊆ {5, 6, 7, 8}

{5, 6, 7, 8} ⊂ {5, 6, 7, 8}

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