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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.

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


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 ≠ ∅

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

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.

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

Q has 3 elements
Number of subsets = 23 = 8
Number of proper subsets = 7

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

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

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