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More Lessons On Sets
Universal Set
Venn Diagrams
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.
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:
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.
How do we find the number of subsets a set has? How many subsets does any given set has? In this video we go over some example problems, calculating the number of subsets a set has.
If a set has n elements then it has 2^{n} 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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