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.
- c ∈ S
- b ∈ S
- d ∈ S
- c ∈ S
- a ∈ S
- 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.
| # | Binary | Use if 1 | Subset |
|---|---|---|---|
| 0 | 000000 | {} | |
| 1 | 000001 | {f} | |
| 2 | 000010 | {a} | |
| 3 | 000011 | {a,f} | |
| 4 | 000100 | {c} | |
| 5 | 000101 | {c,f} | |
| 6 | 000110 | {c,a} | |
| 7 | 000111 | {c,a,f} | |
| 8 | 001000 | {d} | |
| 9 | 001001 | {d,f} | |
| 10 | 001010 | {d,a} | |
| 11 | 001011 | {d,a,f} | |
| 12 | 001100 | {d,c} | |
| 13 | 001101 | {d,c,f} | |
| 14 | 001110 | {d,c,a} | |
| 15 | 001111 | {d,c,a,f} | |
| 16 | 010000 | {b} | |
| 17 | 010001 | {b,f} | |
| 18 | 010010 | {b,a} | |
| 19 | 010011 | {b,a,f} | |
| 20 | 010100 | {b,c} | |
| 21 | 010101 | {b,c,f} | |
| 22 | 010110 | {b,c,a} | |
| 23 | 010111 | {b,c,a,f} | |
| 24 | 011000 | {b,d} | |
| 25 | 011001 | {b,d,f} | |
| 26 | 011010 | {b,d,a} | |
| 27 | 011011 | {b,d,a,f} | |
| 28 | 011100 | {b,d,c} | |
| 29 | 011101 | {b,d,c,f} | |
| 30 | 011110 | {b,d,c,a} | |
| 31 | 011111 | {b,d,c,a,f} | |
| 32 | 100000 | c, | {c} |
| 33 | 100001 | c, | {c,f} |
| 34 | 100010 | c, | {c,a} |
| 35 | 100011 | c, | {c,a,f} |
| 36 | 100100 | c, | {c,c} |
| 37 | 100101 | c, | {c,c,f} |
| 38 | 100110 | c, | {c,c,a} |
| 39 | 100111 | c, | {c,c,a,f} |
| 40 | 101000 | c, | {c,d} |
| 41 | 101001 | c, | {c,d,f} |
| 42 | 101010 | c, | {c,d,a} |
| 43 | 101011 | c, | {c,d,a,f} |
| 44 | 101100 | c, | {c,d,c} |
| 45 | 101101 | c, | {c,d,c,f} |
| 46 | 101110 | c, | {c,d,c,a} |
| 47 | 101111 | c, | {c,d,c,a,f} |
| 48 | 110000 | c,b, | {c,b} |
| 49 | 110001 | c,b, | {c,b,f} |
| 50 | 110010 | c,b, | {c,b,a} |
| 51 | 110011 | c,b, | {c,b,a,f} |
| 52 | 110100 | c,b, | {c,b,c} |
| 53 | 110101 | c,b, | {c,b,c,f} |
| 54 | 110110 | c,b, | {c,b,c,a} |
| 55 | 110111 | c,b, | {c,b,c,a,f} |
| 56 | 111000 | c,b,d, | {c,b,d} |
| 57 | 111001 | c,b,d, | {c,b,d,f} |
| 58 | 111010 | c,b,d, | {c,b,d,a} |
| 59 | 111011 | c,b,d, | {c,b,d,a,f} |
| 60 | 111100 | c,b,d,c, | {c,b,d,c} |
| 61 | 111101 | c,b,d,c, | {c,b,d,c,f} |
| 62 | 111110 | c,b,d,c,a, | {c,b,d,c,a} |
| 63 | 111111 | c,b,d,c,a,f | {c,b,d,c,a,f} |
List our Power Set P in notation form:
You have 2 free calculationss remaining
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})
What is the Answer?
How does the Power Sets and Set Partitions Calculator work?
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What 1 formula is used for the Power Sets and Set Partitions Calculator?
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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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