# Power Sets and Set Partitions Calculator

<-- Enter Set

Display the elements, cardinality, and power set for S = {1,2,3,4,5}

Determine the cardinality of set S denoted as |S|:
The cardinality of a set is the number of items contained in the set.
|S| = 5

List the elements of S
1 ∈ S
2 ∈ S
3 ∈ S
4 ∈ S
5 ∈ S

Determine the power set of S, denoted as P:
The power set is the set of all subsets of S including S and the empty set
Since S contains 5 terms, our Power Set should contain 25 = 32 items
Each subset term can be written using binary expansion representation starting at 0 through 32 - 1 = 31

Term NumberBinary Term1 = Use, 0 = IgnoreSubset
0000001,2,3,4,5{}
1000011,2,3,4,5{5}
2000101,2,3,4,5{4}
3000111,2,3,4,5{4,5}
4001001,2,3,4,5{3}
5001011,2,3,4,5{3,5}
6001101,2,3,4,5{3,4}
7001111,2,3,4,5{3,4,5}
8010001,2,3,4,5{2}
9010011,2,3,4,5{2,5}
10010101,2,3,4,5{2,4}
11010111,2,3,4,5{2,4,5}
12011001,2,3,4,5{2,3}
13011011,2,3,4,5{2,3,5}
14011101,2,3,4,5{2,3,4}
15011111,2,3,4,5{2,3,4,5}
16100001,2,3,4,5{1}
17100011,2,3,4,5{1,5}
18100101,2,3,4,5{1,4}
19100111,2,3,4,5{1,4,5}
20101001,2,3,4,5{1,3}
21101011,2,3,4,5{1,3,5}
22101101,2,3,4,5{1,3,4}
23101111,2,3,4,5{1,3,4,5}
24110001,2,3,4,5{1,2}
25110011,2,3,4,5{1,2,5}
26110101,2,3,4,5{1,2,4}
27110111,2,3,4,5{1,2,4,5}
28111001,2,3,4,5{1,2,3}
29111011,2,3,4,5{1,2,3,5}
30111101,2,3,4,5{1,2,3,4}
31111111,2,3,4,5{1,2,3,4,5}

List our Power Set P in notation form:
P = {{}, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}}

Show the 56 Set Partitions
Partition 1
{4,5},{1,2,3}
Partition 2
{4,5},{1,2,3}
Partition 3
{4,5},{1,2,3}
Partition 4
{3,5},
Partition 5
{3,5},
Partition 6
{3,5},
Partition 7
{3,4},
Partition 8
{3,4},
Partition 9
{3,4},
Partition 10
{3,4,5},{1,2}
Partition 11
{3,4,5},{1,2}
Partition 12
{2,5},{1,2,3}
Partition 13
{2,5},{1,2,3}
Partition 14
{2,5},{1,2,3}
Partition 15
{2,4},{1,2,3}
Partition 16
{2,4},{1,2,3}
Partition 17
{2,4},{1,2,3}
Partition 18
{2,4,5},
Partition 19
{2,4,5},
Partition 20
{2,3},
Partition 21
{2,3},
Partition 22
{2,3},
Partition 23
{2,3,5},
Partition 24
{2,3,5},
Partition 25
{2,3,4},
Partition 26
{2,3,4},
Partition 27
{2,3,4,5},{1}
Partition 28
{1,5},{1,2,3}
Partition 29
{1,5},{1,2,3}
Partition 30
{1,5},{1,2,3}
Partition 31
{1,4},{1,2,3}
Partition 32
{1,4},{1,2,3}
Partition 33
{1,4},{1,2,3}
Partition 34
{1,4,5},{1,2}
Partition 35
{1,4,5},{1,2}
Partition 36
{1,3},
Partition 37
{1,3},
Partition 38
{1,3},
Partition 39
{1,3,5},{1,2}
Partition 40
{1,3,5},{1,2}
Partition 41
{1,3,4},{1,2}
Partition 42
{1,3,4},{1,2}
Partition 43
{1,3,4,5},
Partition 44
{1,2},{1,2,3}
Partition 45
{1,2},{1,2,3}
Partition 46
{1,2},{1,2,3}
Partition 47
{1,2,5},
Partition 48
{1,2,5},
Partition 49
{1,2,4},
Partition 50
{1,2,4},
Partition 51
{1,2,4,5},
Partition 52
{1,2,3},
Partition 53
{1,2,3},
Partition 54
{1,2,3,5},
Partition 55
{1,2,3,4},
Partition 56
{{1},{2},{3},{4},{5})