For set S = {a,b,c,d,e}, show:
Elements, cardinality, and power set
List the elements of S
Elements = set objects
Use the ∈ symbol.
- a ∈ S
- b ∈ S
- c ∈ S
- d ∈ S
- e ∈ S
Cardinality of set S → |S|:
Cardinality = Number of set elements.
Since the set S contains 5 elements
|S| = 5
Determine the power set P:
Power set = Set of all subsets of S
including S and ∅.
Calculate power set subsets
S contains 5 terms
Power Set contains 25 = 32 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 | 00000 | {} | |
| 1 | 00001 | {e} | |
| 2 | 00010 | {d} | |
| 3 | 00011 | {d,e} | |
| 4 | 00100 | {c} | |
| 5 | 00101 | {c,e} | |
| 6 | 00110 | {c,d} | |
| 7 | 00111 | {c,d,e} | |
| 8 | 01000 | {b} | |
| 9 | 01001 | {b,e} | |
| 10 | 01010 | {b,d} | |
| 11 | 01011 | {b,d,e} | |
| 12 | 01100 | {b,c} | |
| 13 | 01101 | {b,c,e} | |
| 14 | 01110 | {b,c,d} | |
| 15 | 01111 | {b,c,d,e} | |
| 16 | 10000 | a, | {a} |
| 17 | 10001 | a, | {a,e} |
| 18 | 10010 | a, | {a,d} |
| 19 | 10011 | a, | {a,d,e} |
| 20 | 10100 | a, | {a,c} |
| 21 | 10101 | a, | {a,c,e} |
| 22 | 10110 | a, | {a,c,d} |
| 23 | 10111 | a, | {a,c,d,e} |
| 24 | 11000 | a,b, | {a,b} |
| 25 | 11001 | a,b, | {a,b,e} |
| 26 | 11010 | a,b, | {a,b,d} |
| 27 | 11011 | a,b, | {a,b,d,e} |
| 28 | 11100 | a,b,c, | {a,b,c} |
| 29 | 11101 | a,b,c, | {a,b,c,e} |
| 30 | 11110 | a,b,c,d, | {a,b,c,d} |
| 31 | 11111 | a,b,c,d,e | {a,b,c,d,e} |
List our Power Set P in notation form:
Partition 1
{d,e},{a,b,c}
Partition 2
{d,e},{a,b,c}
Partition 3
{d,e},{a,b,c}
Partition 4
{c,e},
Partition 5
{c,e},
Partition 6
{c,e},
Partition 7
{c,d},
Partition 8
{c,d},
Partition 9
{c,d},
Partition 10
{c,d,e},{a,b}
Partition 11
{c,d,e},{a,b}
Partition 12
{b,e},{a,b,c}
Partition 13
{b,e},{a,b,c}
Partition 14
{b,e},{a,b,c}
Partition 15
{b,d},{a,b,c}
Partition 16
{b,d},{a,b,c}
Partition 17
{b,d},{a,b,c}
Partition 18
{b,d,e},
Partition 19
{b,d,e},
Partition 20
{b,c},
Partition 21
{b,c},
Partition 22
{b,c},
Partition 23
{b,c,e},
Partition 24
{b,c,e},
Partition 25
{b,c,d},
Partition 26
{b,c,d},
Partition 27
{b,c,d,e},{a}
Partition 28
{a,e},{a,b,c}
Partition 29
{a,e},{a,b,c}
Partition 30
{a,e},{a,b,c}
Partition 31
{a,d},{a,b,c}
Partition 32
{a,d},{a,b,c}
Partition 33
{a,d},{a,b,c}
Partition 34
{a,d,e},{a,b}
Partition 35
{a,d,e},{a,b}
Partition 36
{a,c},
Partition 37
{a,c},
Partition 38
{a,c},
Partition 39
{a,c,e},{a,b}
Partition 40
{a,c,e},{a,b}
Partition 41
{a,c,d},{a,b}
Partition 42
{a,c,d},{a,b}
Partition 43
{a,c,d,e},
Partition 44
{a,b},{a,b,c}
Partition 45
{a,b},{a,b,c}
Partition 46
{a,b},{a,b,c}
Partition 47
{a,b,e},
Partition 48
{a,b,e},
Partition 49
{a,b,d},
Partition 50
{a,b,d},
Partition 51
{a,b,d,e},
Partition 52
{a,b,c},
Partition 53
{a,b,c},
Partition 54
{a,b,c,e},
Partition 55
{a,b,c,d},
Partition 56
{{a},{b},{c},{d},{e})
What is the Answer?
How does the Power Sets and Set Partitions Calculator work?
This calculator has 1 input.
What 1 formula is used for the Power Sets and Set Partitions Calculator?
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
