Power Sets and Set Partitions Calculator

For set S = {1,5,2,3}, show:

Elements, cardinality, and power set

List the elements of S

Elements = set objects
Use the ∈ symbol.

1 ∈ S
5 ∈ S
2 ∈ S
3 ∈ S

Cardinality of set S → |S|:

Cardinality = Number of set elements.

Since the set S contains 4 elements

|S| = 4

Determine the power set P:

Power set = Set of all subsets of S
including S and ∅.

Calculate power set subsets

S contains 4 terms
Power Set contains 2⁴ = 16 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	0000	1,5,2,3	{}
1	0001	1,5,2,3	{3}
2	0010	1,5,2,3	{2}
3	0011	1,5,2,3	{2,3}
4	0100	1,5,2,3	{5}
5	0101	1,5,2,3	{5,3}
6	0110	1,5,2,3	{5,2}
7	0111	1,5,2,3	{5,2,3}
8	1000	1,5,2,3	{1}
9	1001	1,5,2,3	{1,3}
10	1010	1,5,2,3	{1,2}
11	1011	1,5,2,3	{1,2,3}
12	1100	1,5,2,3	{1,5}
13	1101	1,5,2,3	{1,5,3}
14	1110	1,5,2,3	{1,5,2}
15	1111	1,5,2,3	{1,5,2,3}

List our Power Set P in notation form:

P = {{}, {1}, {2}, {3}, {5}, {1,2}, {1,3}, {1,5}, {2,3}, {5,2}, {5,3}, {1,2,3}, {1,5,2}, {1,5,3}, {5,2,3}, {1,5,2,3}}

You have 1 free calculations remaining

Partition 1

{2,3},{1,5}

Partition 2

{2,3},{1,5}

Partition 3

{5,3},

Partition 4

{5,3},

Partition 5

{5,2},

Partition 6

{5,2},

Partition 7

{5,2,3},{1}

Partition 8

{1,3},{1,5}

Partition 9

{1,3},{1,5}

Partition 10

{1,2},{1,5}

Partition 11

{1,2},{1,5}

Partition 12

{1,2,3},

Partition 13

{1,5},

Partition 14

{1,5},

Partition 15

{1,5,3},

Partition 16

{1,5,2},

Partition 17

{{1},{5},{2},{3})

What is the Answer?

P = {{}, {1}, {2}, {3}, {5}, {1,2}, {1,3}, {1,5}, {2,3}, {5,2}, {5,3}, {1,2,3}, {1,5,2}, {1,5,3}, {5,2,3}, {1,5,2,3}}

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