# Power Sets and Set Partitions Calculator

## Enter Set

For set S = {-6,4,3,-9,2,8}, show:
Elements, cardinality, and power set

## List the elements of S

Elements = set objects
Use the ∈ symbol.
1. -6 ∈ S
2. 4 ∈ S
3. 3 ∈ S
4. -9 ∈ S
5. 2 ∈ S
6. 8 ∈ S

## Cardinality of set S → |S|:

Cardinality = Number of set elements.
Since the set S contains 6 elements
|S| = 6

## Determine the power set P:

Power set = Set of all subsets of S
including S and ∅.

## Calculate power set subsets

S contains 6 terms
Power Set contains 26 = 64 items

## Build subsets of P

The subset A of a set B is
A set where all elements of A are in B.
#BinaryUse if 1Subset
0000000-6,4,3,-9,2,8{}
1000001-6,4,3,-9,2,8{8}
2000010-6,4,3,-9,2,8{2}
3000011-6,4,3,-9,2,8{2,8}
4000100-6,4,3,-9,2,8{-9}
5000101-6,4,3,-9,2,8{-9,8}
6000110-6,4,3,-9,2,8{-9,2}
7000111-6,4,3,-9,2,8{-9,2,8}
8001000-6,4,3,-9,2,8{3}
9001001-6,4,3,-9,2,8{3,8}
10001010-6,4,3,-9,2,8{3,2}
11001011-6,4,3,-9,2,8{3,2,8}
12001100-6,4,3,-9,2,8{3,-9}
13001101-6,4,3,-9,2,8{3,-9,8}
14001110-6,4,3,-9,2,8{3,-9,2}
15001111-6,4,3,-9,2,8{3,-9,2,8}
16010000-6,4,3,-9,2,8{4}
17010001-6,4,3,-9,2,8{4,8}
18010010-6,4,3,-9,2,8{4,2}
19010011-6,4,3,-9,2,8{4,2,8}
20010100-6,4,3,-9,2,8{4,-9}
21010101-6,4,3,-9,2,8{4,-9,8}
22010110-6,4,3,-9,2,8{4,-9,2}
23010111-6,4,3,-9,2,8{4,-9,2,8}
24011000-6,4,3,-9,2,8{4,3}
25011001-6,4,3,-9,2,8{4,3,8}
26011010-6,4,3,-9,2,8{4,3,2}
27011011-6,4,3,-9,2,8{4,3,2,8}
28011100-6,4,3,-9,2,8{4,3,-9}
29011101-6,4,3,-9,2,8{4,3,-9,8}
30011110-6,4,3,-9,2,8{4,3,-9,2}
31011111-6,4,3,-9,2,8{4,3,-9,2,8}
32100000-6,4,3,-9,2,8{-6}
33100001-6,4,3,-9,2,8{-6,8}
34100010-6,4,3,-9,2,8{-6,2}
35100011-6,4,3,-9,2,8{-6,2,8}
36100100-6,4,3,-9,2,8{-6,-9}
37100101-6,4,3,-9,2,8{-6,-9,8}
38100110-6,4,3,-9,2,8{-6,-9,2}
39100111-6,4,3,-9,2,8{-6,-9,2,8}
40101000-6,4,3,-9,2,8{-6,3}
41101001-6,4,3,-9,2,8{-6,3,8}
42101010-6,4,3,-9,2,8{-6,3,2}
43101011-6,4,3,-9,2,8{-6,3,2,8}
44101100-6,4,3,-9,2,8{-6,3,-9}
45101101-6,4,3,-9,2,8{-6,3,-9,8}
46101110-6,4,3,-9,2,8{-6,3,-9,2}
47101111-6,4,3,-9,2,8{-6,3,-9,2,8}
48110000-6,4,3,-9,2,8{-6,4}
49110001-6,4,3,-9,2,8{-6,4,8}
50110010-6,4,3,-9,2,8{-6,4,2}
51110011-6,4,3,-9,2,8{-6,4,2,8}
52110100-6,4,3,-9,2,8{-6,4,-9}
53110101-6,4,3,-9,2,8{-6,4,-9,8}
54110110-6,4,3,-9,2,8{-6,4,-9,2}
55110111-6,4,3,-9,2,8{-6,4,-9,2,8}
56111000-6,4,3,-9,2,8{-6,4,3}
57111001-6,4,3,-9,2,8{-6,4,3,8}
58111010-6,4,3,-9,2,8{-6,4,3,2}
59111011-6,4,3,-9,2,8{-6,4,3,2,8}
60111100-6,4,3,-9,2,8{-6,4,3,-9}
61111101-6,4,3,-9,2,8{-6,4,3,-9,8}
62111110-6,4,3,-9,2,8{-6,4,3,-9,2}
63111111-6,4,3,-9,2,8{-6,4,3,-9,2,8}

## Partition 1

{2,8},{-6,4,3,-9}

## Partition 2

{2,8},{-6,4,3,-9}

## Partition 3

{2,8},{-6,4,3,-9}

## Partition 4

{2,8},{-6,4,3,-9}

{-9,8},

{-9,8},

{-9,8},

{-9,8},

{-9,2},

{-9,2},

{-9,2},

{-9,2},

## Partition 13

{-9,2,8},{-6,4,3}

## Partition 14

{-9,2,8},{-6,4,3}

## Partition 15

{-9,2,8},{-6,4,3}

## Partition 16

{3,8},{-6,4,3,-9}

## Partition 17

{3,8},{-6,4,3,-9}

## Partition 18

{3,8},{-6,4,3,-9}

## Partition 19

{3,8},{-6,4,3,-9}

## Partition 20

{3,2},{-6,4,3,-9}

## Partition 21

{3,2},{-6,4,3,-9}

## Partition 22

{3,2},{-6,4,3,-9}

## Partition 23

{3,2},{-6,4,3,-9}

{3,2,8},

{3,2,8},

{3,2,8},

{3,-9},

{3,-9},

{3,-9},

{3,-9},

{3,-9,8},

{3,-9,8},

{3,-9,8},

{3,-9,2},

{3,-9,2},

{3,-9,2},

## Partition 37

{3,-9,2,8},{-6,4}

## Partition 38

{3,-9,2,8},{-6,4}

## Partition 39

{4,8},{-6,4,3,-9}

## Partition 40

{4,8},{-6,4,3,-9}

## Partition 41

{4,8},{-6,4,3,-9}

## Partition 42

{4,8},{-6,4,3,-9}

## Partition 43

{4,2},{-6,4,3,-9}

## Partition 44

{4,2},{-6,4,3,-9}

## Partition 45

{4,2},{-6,4,3,-9}

## Partition 46

{4,2},{-6,4,3,-9}

{4,2,8},{-6,4,3}

{4,2,8},{-6,4,3}

{4,2,8},{-6,4,3}

{4,-9},

{4,-9},

{4,-9},

{4,-9},

## Partition 54

{4,-9,8},{-6,4,3}

## Partition 55

{4,-9,8},{-6,4,3}

## Partition 56

{4,-9,8},{-6,4,3}

## Partition 57

{4,-9,2},{-6,4,3}

## Partition 58

{4,-9,2},{-6,4,3}

## Partition 59

{4,-9,2},{-6,4,3}

{4,-9,2,8},

{4,-9,2,8},

## Partition 62

{4,3},{-6,4,3,-9}

## Partition 63

{4,3},{-6,4,3,-9}

## Partition 64

{4,3},{-6,4,3,-9}

## Partition 65

{4,3},{-6,4,3,-9}

{4,3,8},

{4,3,8},

{4,3,8},

{4,3,2},

{4,3,2},

{4,3,2},

{4,3,2,8},

{4,3,2,8},

{4,3,-9},

{4,3,-9},

{4,3,-9},

{4,3,-9,8},

{4,3,-9,8},

{4,3,-9,2},

{4,3,-9,2},

## Partition 81

{4,3,-9,2,8},{-6}

## Partition 82

{-6,8},{-6,4,3,-9}

## Partition 83

{-6,8},{-6,4,3,-9}

## Partition 84

{-6,8},{-6,4,3,-9}

## Partition 85

{-6,8},{-6,4,3,-9}

## Partition 86

{-6,2},{-6,4,3,-9}

## Partition 87

{-6,2},{-6,4,3,-9}

## Partition 88

{-6,2},{-6,4,3,-9}

## Partition 89

{-6,2},{-6,4,3,-9}

## Partition 90

{-6,2,8},{-6,4,3}

## Partition 91

{-6,2,8},{-6,4,3}

## Partition 92

{-6,2,8},{-6,4,3}

{-6,-9},

{-6,-9},

{-6,-9},

{-6,-9},

## Partition 97

{-6,-9,8},{-6,4,3}

## Partition 98

{-6,-9,8},{-6,4,3}

## Partition 99

{-6,-9,8},{-6,4,3}

## Partition 100

{-6,-9,2},{-6,4,3}

## Partition 101

{-6,-9,2},{-6,4,3}

## Partition 102

{-6,-9,2},{-6,4,3}

## Partition 103

{-6,-9,2,8},{-6,4}

## Partition 104

{-6,-9,2,8},{-6,4}

## Partition 105

{-6,3},{-6,4,3,-9}

## Partition 106

{-6,3},{-6,4,3,-9}

## Partition 107

{-6,3},{-6,4,3,-9}

## Partition 108

{-6,3},{-6,4,3,-9}

{-6,3,8},

{-6,3,8},

{-6,3,8},

{-6,3,2},

{-6,3,2},

{-6,3,2},

## Partition 115

{-6,3,2,8},{-6,4}

## Partition 116

{-6,3,2,8},{-6,4}

{-6,3,-9},

{-6,3,-9},

{-6,3,-9},

## Partition 120

{-6,3,-9,8},{-6,4}

## Partition 121

{-6,3,-9,8},{-6,4}

## Partition 122

{-6,3,-9,2},{-6,4}

## Partition 123

{-6,3,-9,2},{-6,4}

{-6,3,-9,2,8},

## Partition 125

{-6,4},{-6,4,3,-9}

## Partition 126

{-6,4},{-6,4,3,-9}

## Partition 127

{-6,4},{-6,4,3,-9}

## Partition 128

{-6,4},{-6,4,3,-9}

## Partition 129

{-6,4,8},{-6,4,3}

## Partition 130

{-6,4,8},{-6,4,3}

## Partition 131

{-6,4,8},{-6,4,3}

## Partition 132

{-6,4,2},{-6,4,3}

## Partition 133

{-6,4,2},{-6,4,3}

## Partition 134

{-6,4,2},{-6,4,3}

{-6,4,2,8},

{-6,4,2,8},

## Partition 137

{-6,4,-9},{-6,4,3}

## Partition 138

{-6,4,-9},{-6,4,3}

## Partition 139

{-6,4,-9},{-6,4,3}

{-6,4,-9,8},

{-6,4,-9,8},

{-6,4,-9,2},

{-6,4,-9,2},

{-6,4,-9,2,8},

{-6,4,3},

{-6,4,3},

{-6,4,3},

{-6,4,3,8},

{-6,4,3,8},

{-6,4,3,2},

{-6,4,3,2},

{-6,4,3,2,8},

{-6,4,3,-9},

{-6,4,3,-9},

{-6,4,3,-9,8},

{-6,4,3,-9,2},

## Partition 157

{{-6},{4},{3},{-9},{2},{8})

P = {{}, {-9}, {-6}, {2}, {3}, {4}, {8}, {-6,-9}, {-6,2}, {-6,3}, {-6,4}, {-6,8}, {-9,2}, {-9,8}, {2,8}, {3,-9}, {3,2}, {3,8}, {4,-9}, {4,2}, {4,3}, {4,8}, {-6,-9,2}, {-6,-9,8}, {-6,2,8}, {-6,3,-9}, {-6,3,2}, {-6,3,8}, {-6,4,-9}, {-6,4,2}, {-6,4,3}, {-6,4,8}, {-9,2,8}, {3,-9,2}, {3,-9,8}, {3,2,8}, {4,-9,2}, {4,-9,8}, {4,2,8}, {4,3,-9}, {4,3,2}, {4,3,8}, {-6,-9,2,8}, {-6,3,-9,2}, {-6,3,-9,8}, {-6,3,2,8}, {-6,4,-9,2}, {-6,4,-9,8}, {-6,4,2,8}, {-6,4,3,-9}, {-6,4,3,2}, {-6,4,3,8}, {3,-9,2,8}, {4,-9,2,8}, {4,3,-9,2}, {4,3,-9,8}, {4,3,2,8}, {-6,3,-9,2,8}, {-6,4,-9,2,8}, {-6,4,3,-9,2}, {-6,4,3,-9,8}, {-6,4,3,2,8}, {4,3,-9,2,8}, {-6,4,3,-9,2,8}}

### How does the Power Sets and Set Partitions Calculator work?

Free Power Sets and Set Partitions Calculator - Given a set S, this calculator will determine the power set for S and all the partitions of a set.
This calculator has 1 input.

### What 1 formula is used for the Power Sets and Set Partitions Calculator?

1. The power set P is the set of all subsets of S including S and the empty set ∅.

For more math formulas, check out our Formula Dossier

### What 7 concepts are covered in the Power Sets and Set Partitions Calculator?

element
an element (or member) of a set is any one of the distinct objects that belong to that set. In chemistry, any substance that cannot be decomposed into simpler substances by ordinary chemical processes.
empty set
The set with no elements
notation
An expression made up of symbols for representing operations, unspecified numbers, relations and any other mathematical objects
partition
a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.
power sets and set partitions
set
a collection of different things; a set contains elements or members, which can be mathematical objects of any kind
subset
A is a subset of B if all elements of the set A are elements of the set B