Set Theory: Subsets

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.

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, CB. 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 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 2n subsets.
If set A has n elements, it has 2n - 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 = 23 = 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} Subsets of the Real Number System
Brief description of the set of real numbers and its subsets: naturals, wholes, integers, rationals, and irrationals.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.