For set S = {1,5,2,3}, show:
Elements, cardinality, and power set
List the elements of S
Elements = set objects
Use the ∈ symbol.
- 1 ∈ S
- 5 ∈ S
- 2 ∈ S
- 3 ∈ S
Cardinality of set S → |S|:
Cardinality = Number of set elements.
Since the set S contains 4 elements
|S| = 4
Determine the power set P:
Power set = Set of all subsets of S
including S and ∅.
Calculate power set subsets
S contains 4 terms
Power Set contains 24 = 16 items
Build subsets of P
The subset A of a set B is
A set where all elements of A are in B.
| # | Binary | Use if 1 | Subset |
|---|---|---|---|
| 0 | 0000 | {} | |
| 1 | 0001 | {3} | |
| 2 | 0010 | {2} | |
| 3 | 0011 | {2,3} | |
| 4 | 0100 | {5} | |
| 5 | 0101 | {5,3} | |
| 6 | 0110 | {5,2} | |
| 7 | 0111 | {5,2,3} | |
| 8 | 1000 | 1, | {1} |
| 9 | 1001 | 1, | {1,3} |
| 10 | 1010 | 1, | {1,2} |
| 11 | 1011 | 1, | {1,2,3} |
| 12 | 1100 | 1,5, | {1,5} |
| 13 | 1101 | 1,5, | {1,5,3} |
| 14 | 1110 | 1,5,2, | {1,5,2} |
| 15 | 1111 | 1,5,2,3 | {1,5,2,3} |
List our Power Set P in notation form:
P = {{}, {1}, {2}, {3}, {5}, {1,2}, {1,3}, {1,5}, {2,3}, {5,2}, {5,3}, {1,2,3}, {1,5,2}, {1,5,3}, {5,2,3}, {1,5,2,3}}
Partition 1
{2,3},{1,5}
Partition 2
{2,3},{1,5}
Partition 3
{5,3},
Partition 4
{5,3},
Partition 5
{5,2},
Partition 6
{5,2},
Partition 7
{5,2,3},{1}
Partition 8
{1,3},{1,5}
Partition 9
{1,3},{1,5}
Partition 10
{1,2},{1,5}
Partition 11
{1,2},{1,5}
Partition 12
{1,2,3},
Partition 13
{1,5},
Partition 14
{1,5},
Partition 15
{1,5,3},
Partition 16
{1,5,2},
Partition 17
{{1},{5},{2},{3})
