Enter Set

For set S = {c,b,d,c,a,f}, show:

Elements, cardinality, and power set

List the elements of S

Elements = set objects
Use the ∈ symbol.

  1. c ∈ S
  2. b ∈ S
  3. d ∈ S
  4. c ∈ S
  5. a ∈ S
  6. f ∈ 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
0000000c,b,d,c,a,f{}
1000001c,b,d,c,a,f{f}
2000010c,b,d,c,a,f{a}
3000011c,b,d,c,a,f{a,f}
4000100c,b,d,c,a,f{c}
5000101c,b,d,c,a,f{c,f}
6000110c,b,d,c,a,f{c,a}
7000111c,b,d,c,a,f{c,a,f}
8001000c,b,d,c,a,f{d}
9001001c,b,d,c,a,f{d,f}
10001010c,b,d,c,a,f{d,a}
11001011c,b,d,c,a,f{d,a,f}
12001100c,b,d,c,a,f{d,c}
13001101c,b,d,c,a,f{d,c,f}
14001110c,b,d,c,a,f{d,c,a}
15001111c,b,d,c,a,f{d,c,a,f}
16010000c,b,d,c,a,f{b}
17010001c,b,d,c,a,f{b,f}
18010010c,b,d,c,a,f{b,a}
19010011c,b,d,c,a,f{b,a,f}
20010100c,b,d,c,a,f{b,c}
21010101c,b,d,c,a,f{b,c,f}
22010110c,b,d,c,a,f{b,c,a}
23010111c,b,d,c,a,f{b,c,a,f}
24011000c,b,d,c,a,f{b,d}
25011001c,b,d,c,a,f{b,d,f}
26011010c,b,d,c,a,f{b,d,a}
27011011c,b,d,c,a,f{b,d,a,f}
28011100c,b,d,c,a,f{b,d,c}
29011101c,b,d,c,a,f{b,d,c,f}
30011110c,b,d,c,a,f{b,d,c,a}
31011111c,b,d,c,a,f{b,d,c,a,f}
32100000c,b,d,c,a,f{c}
33100001c,b,d,c,a,f{c,f}
34100010c,b,d,c,a,f{c,a}
35100011c,b,d,c,a,f{c,a,f}
36100100c,b,d,c,a,f{c,c}
37100101c,b,d,c,a,f{c,c,f}
38100110c,b,d,c,a,f{c,c,a}
39100111c,b,d,c,a,f{c,c,a,f}
40101000c,b,d,c,a,f{c,d}
41101001c,b,d,c,a,f{c,d,f}
42101010c,b,d,c,a,f{c,d,a}
43101011c,b,d,c,a,f{c,d,a,f}
44101100c,b,d,c,a,f{c,d,c}
45101101c,b,d,c,a,f{c,d,c,f}
46101110c,b,d,c,a,f{c,d,c,a}
47101111c,b,d,c,a,f{c,d,c,a,f}
48110000c,b,d,c,a,f{c,b}
49110001c,b,d,c,a,f{c,b,f}
50110010c,b,d,c,a,f{c,b,a}
51110011c,b,d,c,a,f{c,b,a,f}
52110100c,b,d,c,a,f{c,b,c}
53110101c,b,d,c,a,f{c,b,c,f}
54110110c,b,d,c,a,f{c,b,c,a}
55110111c,b,d,c,a,f{c,b,c,a,f}
56111000c,b,d,c,a,f{c,b,d}
57111001c,b,d,c,a,f{c,b,d,f}
58111010c,b,d,c,a,f{c,b,d,a}
59111011c,b,d,c,a,f{c,b,d,a,f}
60111100c,b,d,c,a,f{c,b,d,c}
61111101c,b,d,c,a,f{c,b,d,c,f}
62111110c,b,d,c,a,f{c,b,d,c,a}
63111111c,b,d,c,a,f{c,b,d,c,a,f}

List our Power Set P in notation form:


P = {{}, {a}, {b}, {c}, {c}, {d}, {f}, {a,f}, {b,a}, {b,c}, {b,d}, {b,f}, {c,a}, {c,a}, {c,b}, {c,c}, {c,d}, {c,f}, {c,f}, {d,a}, {d,c}, {d,f}, {b,a,f}, {b,c,a}, {b,c,f}, {b,d,a}, {b,d,c}, {b,d,f}, {c,a,f}, {c,a,f}, {c,b,a}, {c,b,c}, {c,b,d}, {c,b,f}, {c,c,a}, {c,c,f}, {c,d,a}, {c,d,c}, {c,d,f}, {d,a,f}, {d,c,a}, {d,c,f}, {b,c,a,f}, {b,d,a,f}, {b,d,c,a}, {b,d,c,f}, {c,b,a,f}, {c,b,c,a}, {c,b,c,f}, {c,b,d,a}, {c,b,d,c}, {c,b,d,f}, {c,c,a,f}, {c,d,a,f}, {c,d,c,a}, {c,d,c,f}, {d,c,a,f}, {b,d,c,a,f}, {c,b,c,a,f}, {c,b,d,a,f}, {c,b,d,c,a}, {c,b,d,c,f}, {c,d,c,a,f}, {c,b,d,c,a,f}}

Partition 1

{a,f},{c,b,d,c}

Partition 2

{a,f},{c,b,d,c}

Partition 3

{a,f},{c,b,d,c}

Partition 4

{a,f},{c,b,d,c}

Partition 5

{c,f},

Partition 6

{c,f},

Partition 7

{c,f},

Partition 8

{c,f},

Partition 9

{c,a},

Partition 10

{c,a},

Partition 11

{c,a},

Partition 12

{c,a},

Partition 13

{c,a,f},{c,b,d}

Partition 14

{c,a,f},{c,b,d}

Partition 15

{c,a,f},{c,b,d}

Partition 16

{d,f},{c,b,d,c}

Partition 17

{d,f},{c,b,d,c}

Partition 18

{d,f},{c,b,d,c}

Partition 19

{d,f},{c,b,d,c}

Partition 20

{d,a},{c,b,d,c}

Partition 21

{d,a},{c,b,d,c}

Partition 22

{d,a},{c,b,d,c}

Partition 23

{d,a},{c,b,d,c}

Partition 24

{d,a,f},

Partition 25

{d,a,f},

Partition 26

{d,a,f},

Partition 27

{d,c},

Partition 28

{d,c},

Partition 29

{d,c},

Partition 30

{d,c},

Partition 31

{d,c,f},

Partition 32

{d,c,f},

Partition 33

{d,c,f},

Partition 34

{d,c,a},

Partition 35

{d,c,a},

Partition 36

{d,c,a},

Partition 37

{d,c,a,f},{c,b}

Partition 38

{d,c,a,f},{c,b}

Partition 39

{b,f},{c,b,d,c}

Partition 40

{b,f},{c,b,d,c}

Partition 41

{b,f},{c,b,d,c}

Partition 42

{b,f},{c,b,d,c}

Partition 43

{b,a},{c,b,d,c}

Partition 44

{b,a},{c,b,d,c}

Partition 45

{b,a},{c,b,d,c}

Partition 46

{b,a},{c,b,d,c}

Partition 47

{b,a,f},{c,b,d}

Partition 48

{b,a,f},{c,b,d}

Partition 49

{b,a,f},{c,b,d}

Partition 50

{b,c},

Partition 51

{b,c},

Partition 52

{b,c},

Partition 53

{b,c},

Partition 54

{b,c,f},{c,b,d}

Partition 55

{b,c,f},{c,b,d}

Partition 56

{b,c,f},{c,b,d}

Partition 57

{b,c,a},{c,b,d}

Partition 58

{b,c,a},{c,b,d}

Partition 59

{b,c,a},{c,b,d}

Partition 60

{b,c,a,f},

Partition 61

{b,c,a,f},

Partition 62

{b,d},{c,b,d,c}

Partition 63

{b,d},{c,b,d,c}

Partition 64

{b,d},{c,b,d,c}

Partition 65

{b,d},{c,b,d,c}

Partition 66

{b,d,f},

Partition 67

{b,d,f},

Partition 68

{b,d,f},

Partition 69

{b,d,a},

Partition 70

{b,d,a},

Partition 71

{b,d,a},

Partition 72

{b,d,a,f},

Partition 73

{b,d,a,f},

Partition 74

{b,d,c},

Partition 75

{b,d,c},

Partition 76

{b,d,c},

Partition 77

{b,d,c,f},

Partition 78

{b,d,c,f},

Partition 79

{b,d,c,a},

Partition 80

{b,d,c,a},

Partition 81

{b,d,c,a,f},

Partition 82

{c,f},

Partition 83

{c,f},

Partition 84

{c,f},

Partition 85

{c,f},

Partition 86

{c,a},

Partition 87

{c,a},

Partition 88

{c,a},

Partition 89

{c,a},

Partition 90

{c,a,f},{c,b,d}

Partition 91

{c,a,f},{c,b,d}

Partition 92

{c,a,f},{c,b,d}

Partition 93

{c,c},

Partition 94

{c,c},

Partition 95

{c,c},

Partition 96

{c,c},

Partition 97

{c,c,f},{c,b,d}

Partition 98

{c,c,f},{c,b,d}

Partition 99

{c,c,f},{c,b,d}

Partition 100

{c,c,a},{c,b,d}

Partition 101

{c,c,a},{c,b,d}

Partition 102

{c,c,a},{c,b,d}

Partition 103

{c,c,a,f},{c,b}

Partition 104

{c,c,a,f},{c,b}

Partition 105

{c,d},

Partition 106

{c,d},

Partition 107

{c,d},

Partition 108

{c,d},

Partition 109

{c,d,f},

Partition 110

{c,d,f},

Partition 111

{c,d,f},

Partition 112

{c,d,a},

Partition 113

{c,d,a},

Partition 114

{c,d,a},

Partition 115

{c,d,a,f},{c,b}

Partition 116

{c,d,a,f},{c,b}

Partition 117

{c,d,c},

Partition 118

{c,d,c},

Partition 119

{c,d,c},

Partition 120

{c,d,c,f},{c,b}

Partition 121

{c,d,c,f},{c,b}

Partition 122

{c,d,c,a},{c,b}

Partition 123

{c,d,c,a},{c,b}

Partition 124

{c,d,c,a,f},

Partition 125

{c,b},

Partition 126

{c,b},

Partition 127

{c,b},

Partition 128

{c,b},

Partition 129

{c,b,f},{c,b,d}

Partition 130

{c,b,f},{c,b,d}

Partition 131

{c,b,f},{c,b,d}

Partition 132

{c,b,a},{c,b,d}

Partition 133

{c,b,a},{c,b,d}

Partition 134

{c,b,a},{c,b,d}

Partition 135

{c,b,a,f},

Partition 136

{c,b,a,f},

Partition 137

{c,b,c},{c,b,d}

Partition 138

{c,b,c},{c,b,d}

Partition 139

{c,b,c},{c,b,d}

Partition 140

{c,b,c,f},

Partition 141

{c,b,c,f},

Partition 142

{c,b,c,a},

Partition 143

{c,b,c,a},

Partition 144

{c,b,c,a,f},

Partition 145

{c,b,d},

Partition 146

{c,b,d},

Partition 147

{c,b,d},

Partition 148

{c,b,d,f},

Partition 149

{c,b,d,f},

Partition 150

{c,b,d,a},

Partition 151

{c,b,d,a},

Partition 152

{c,b,d,a,f},

Partition 153

{c,b,d,c},

Partition 154

{c,b,d,c},

Partition 155

{c,b,d,c,f},

Partition 156

{c,b,d,c,a},

Partition 157

{{c},{b},{d},{c},{a},{f})

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