Subsets
Subsets
Subsets are sort of like nested Russian dolls:
the subset “fits inside” the set.
Subsets
Set A is a subset of set B if all of the members in
set A are also in set B.
set A = { 1, 2, 3 } set B = { 1, 2, 3, 4, 5, 6 }
Set A is a subset of set B
A B
NOT a Subset
Set A is NOT a subset of set B if any member in set
A is not also in set B.
set A = { 1, 2, 9 }
set B = { 1, 2, 3, 4, 5, 6 }
Set A is NOT a subset of set B because of the 9.
A B
Proper Subsets
If set A has fewer elements than set B, it is called a
proper subset and we use

The idea is very similar to the concept of “less than”
‹
Improper Subsets
If they are the same set, it is called an improper
subset and we use  (or we could use =).
By definition, every set is a subset of itself.
A A
Generic Subsets
If we aren’t sure if it’s a proper subset or an improper
subset, we use the generic subset symbol.

That way it is true either if it is proper or improper.
The Empty Set
Also by definition:
The empty set is a subset of every other set.
For every set A:
 A
Listing Subsets
Let’s list all the subsets of the set {1, 2}
The proper subsets: {1}, {2}, and 
The improper subset: {1, 2}
2
A set with 2 elements has 2 or 4 subsets
2
A set with 2 elements has 2 – 1 or 3 proper subsets
Listing Subsets
Let’s list all the subsets of the set {1, 2, 3}
The proper subsets: {1}, {2}, {3}, 
{1, 2}, {1, 3}, {2, 3}
The improper subset: {1, 2, 3}
3
A set with 3 elements has 2 or 8 subsets
3
A set with 3 elements has 2 – 1 or 7 subsets
Number of Subsets
If a set has n elements
The number of subsets will be 2
n
n
The number of proper subsets will be 2 – 1
Vocabulary
Subset
Proper subset
Improper subset
Empty set