1. Section 2.2 Subsets Objectives
Recognize subsets and use the notation .
Determine the number of subsets of a set.
Apply concepts of subsets and equivalent sets to infinite sets.
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2. Subsets Set A is a subset of set B, expressed as A  B,
if every element in set A is also an element in set B.
The notation  means that A is not a subset of B.
A is not a subset of set B if there is at least one element of set A that is not an element of set B.
Every set is a subset of itself.
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3. Universal Set
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4. Example 1 Subsets
Percentage of Tattooed Americans, By Age Group
Age Group
Percent Tattooed
18-24
13%
25-29
32%
30-39
24%
40-49
14%
50-64
10%
65+ 7%
Applying the subset definition to the set of people age in
this table:
Write in Standard Notation Form
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5. Example 1 continued Given: A = {1, 2, 3} B = {1, 2 }
Is A a subset of B?
No. A  B
Is B a subset of A?
Yes. B  A
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6. Example 2 . Write  the blank to form a true statement.
A = { x | x is a person and x lives in San Francisco}
B = { x | x is a person and x lives in California}
A ____B
Solution: A  B
A = { 2, 4, 6, 8}
B = { 2, 8, 4, 6}
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7. Subsets and the Empty Set
The Empty Set as a Subset
For any set B, Ø  B.
For any set B other than the empty set, Ø  B.
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8. The Number of Subsets of a Given Set
Number of Elements
List of All Subsets
Number of Subsets
{ } 1
{a}
{a},{ } 2
{a,b}
{a,b},{a}, {b},{ } 4
{a,b,c} 3
{a,b,c},{a,b}, {a,c},{ b,c },
{a},{b},{c}, { } 8
As we increase the number of elements in the set by one, the number of subsets doubles.
The number of subsets of a set with n elements is 2n.
The number of proper subsets of a set with n elements is 2n – 1.
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9. Example 3 Finding the Number of Subsets
Find the number of subsets and the number of proper subsets.
{a, b, c, d, e }
There are 5 elements so there are 25 = 32 subsets
{ x | x   and 9 ≤ x ≤ 15 }
In roster form, we see that there are 7 elements:
{ 9, 10, 11, 12, 13, 14, 15 }
There are 27 = 128 subsets
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10. Cardinal Numbers of Infinite Sets
Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal number א0 to the set of natural numbers. He used one-to-one correspondences to establish some surprising equivalences between the set of natural numbers and its proper subsets.
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11. Practice
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12. Practice
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