Related Topics: More Lessons on Sets

In these lessons, we will learn

• about subsets and proper subsets

• the formula for the number of subsets

**Subsets and Proper Subsets**

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.

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:

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

∅ ⊆ S

• The empty set is a proper set 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}

** What is the formula for the number of subsets and proper subsets?**

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*

**Subset and Proper Subset**

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.**Find subsets and proper subsets**

Example:

Given the set C = {1, 2, 3}, what are the subsets and proper subsets?**How to distinguish between elements, subsets and proper subsets?**

Examples:

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}

In these lessons, we will learn

• about subsets and proper subsets

• the formula for the number of subsets

If every member of set A is also a member of set B, then A is a

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

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.

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

∅ ⊆ S

• The empty set is a proper set 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

If set A has n elements, it has 2

Example:

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

Q has 3 elements

Number of subsets = 2

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

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?

Examples:

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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