End of Lesson
End of Lesson
The power set of is denoted as where is the set containing all the possible subsets of .
Consider the set . How many subsets does it have? It has four subsets given below.
Putting these four subsets of in one set makes the power set of which has four elements, each of them being a set.
The number of subsets of a set can be found by the relation where is the cardinality of the set. This table shows the relation between the number of sets and cardinality more clearly.
| Cardinality () | Total Subsets () |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
| 10 | 1024 |
Note that the empty set also has one subset which is also an empty set. And a singleton set has two subsets, itself and the empty set.
This table displays the power sets of sets of cardinality from 0 to 5 so that you can see how to construct them intuitively.
| Set | Power Set |
|---|---|
In these next exercises, press enter with input field empty to enter the null set.
Read each statement carefully and decide if it is true or false.
If a set has elements, its power set has elements.
The power set of a finite set is always finite.
The power set of an infinite set is infinite.
is always an element of the power set of any set.
If has 3 elements, has 6 elements.