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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}

List our Power Set P in notation form:

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}}



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}

Partition 5

{-9,8},

Partition 6

{-9,8},

Partition 7

{-9,8},

Partition 8

{-9,8},

Partition 9

{-9,2},

Partition 10

{-9,2},

Partition 11

{-9,2},

Partition 12

{-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}

Partition 24

{3,2,8},

Partition 25

{3,2,8},

Partition 26

{3,2,8},

Partition 27

{3,-9},

Partition 28

{3,-9},

Partition 29

{3,-9},

Partition 30

{3,-9},

Partition 31

{3,-9,8},

Partition 32

{3,-9,8},

Partition 33

{3,-9,8},

Partition 34

{3,-9,2},

Partition 35

{3,-9,2},

Partition 36

{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}

Partition 47

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

Partition 48

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

Partition 49

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

Partition 50

{4,-9},

Partition 51

{4,-9},

Partition 52

{4,-9},

Partition 53

{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}

Partition 60

{4,-9,2,8},

Partition 61

{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}

Partition 66

{4,3,8},

Partition 67

{4,3,8},

Partition 68

{4,3,8},

Partition 69

{4,3,2},

Partition 70

{4,3,2},

Partition 71

{4,3,2},

Partition 72

{4,3,2,8},

Partition 73

{4,3,2,8},

Partition 74

{4,3,-9},

Partition 75

{4,3,-9},

Partition 76

{4,3,-9},

Partition 77

{4,3,-9,8},

Partition 78

{4,3,-9,8},

Partition 79

{4,3,-9,2},

Partition 80

{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}

Partition 93

{-6,-9},

Partition 94

{-6,-9},

Partition 95

{-6,-9},

Partition 96

{-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}

Partition 109

{-6,3,8},

Partition 110

{-6,3,8},

Partition 111

{-6,3,8},

Partition 112

{-6,3,2},

Partition 113

{-6,3,2},

Partition 114

{-6,3,2},

Partition 115

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

Partition 116

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

Partition 117

{-6,3,-9},

Partition 118

{-6,3,-9},

Partition 119

{-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}

Partition 124

{-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}

Partition 135

{-6,4,2,8},

Partition 136

{-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}

Partition 140

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

Partition 141

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

Partition 142

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

Partition 143

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

Partition 144

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

Partition 145

{-6,4,3},

Partition 146

{-6,4,3},

Partition 147

{-6,4,3},

Partition 148

{-6,4,3,8},

Partition 149

{-6,4,3,8},

Partition 150

{-6,4,3,2},

Partition 151

{-6,4,3,2},

Partition 152

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

Partition 153

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

Partition 154

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

Partition 155

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

Partition 156

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

Partition 157

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

Keep Practicing
What is the Answer?
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?

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
Example calculations for the Power Sets and Set Partitions Calculator
Power Sets and Set Partitions Calculator Video

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