# Power Set for S = {c,b,d,c,a,f}

## 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|:

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

## Determine the power set P:

Define 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

A 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}}

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

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

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

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

{c,f},

{c,f},

{c,f},

{c,f},

{c,a},

{c,a},

{c,a},

{c,a},

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

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

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

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

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

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

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

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

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

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

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

{d,a,f},

{d,a,f},

{d,a,f},

{d,c},

{d,c},

{d,c},

{d,c},

{d,c,f},

{d,c,f},

{d,c,f},

{d,c,a},

{d,c,a},

{d,c,a},

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

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

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

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

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

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

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

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

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

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

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

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

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

{b,c},

{b,c},

{b,c},

{b,c},

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

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

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

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

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

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

{b,c,a,f},

{b,c,a,f},

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

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

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

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

{b,d,f},

{b,d,f},

{b,d,f},

{b,d,a},

{b,d,a},

{b,d,a},

{b,d,a,f},

{b,d,a,f},

{b,d,c},

{b,d,c},

{b,d,c},

{b,d,c,f},

{b,d,c,f},

{b,d,c,a},

{b,d,c,a},

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

{c,f},

{c,f},

{c,f},

{c,f},

{c,a},

{c,a},

{c,a},

{c,a},

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

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

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

{c,c},

{c,c},

{c,c},

{c,c},

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

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

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

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

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

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

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

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

{c,d},

{c,d},

{c,d},

{c,d},

{c,d,f},

{c,d,f},

{c,d,f},

{c,d,a},

{c,d,a},

{c,d,a},

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

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

{c,d,c},

{c,d,c},

{c,d,c},

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

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

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

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

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

{c,b},

{c,b},

{c,b},

{c,b},

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

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

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

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

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

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

{c,b,a,f},

{c,b,a,f},

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

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

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

{c,b,c,f},

{c,b,c,f},

{c,b,c,a},

{c,b,c,a},

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

{c,b,d},

{c,b,d},

{c,b,d},

{c,b,d,f},

{c,b,d,f},

{c,b,d,a},

{c,b,d,a},

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

{c,b,d,c},

{c,b,d,c},

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

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

## Partition 157

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

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}}

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

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