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

Elements, cardinality, and power set

Elements = set objects

Use the ∈ symbol.

- a ∈ S
- b ∈ S
- c ∈ S
- d ∈ S
- e ∈ S

Cardinality = Number of set elements.

Since the set S contains 5 elements

|S| = **5**

Power set = Set of all subsets of S

including S and ∅.

S contains 5 terms

Power Set contains 2^{5} = 32 items

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} |

P = **{{}, {a}, {b}, {c}, {d}, {e}, {a,b}, {a,c}, {a,d}, {a,e}, {b,c}, {b,d}, {b,e}, {c,d}, {c,e}, {d,e}, {a,b,c}, {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {a,d,e}, {b,c,d}, {b,c,e}, {b,d,e}, {c,d,e}, {a,b,c,d}, {a,b,c,e}, {a,b,d,e}, {a,c,d,e}, {b,c,d,e}, {a,b,c,d,e}}**

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

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

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

{c,e},

{c,e},

{c,e},

{c,d},

{c,d},

{c,d},

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

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

{b,e},{a,b,c}

{b,e},{a,b,c}

{b,e},{a,b,c}

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

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

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

{b,d,e},

{b,d,e},

{b,c},

{b,c},

{b,c},

{b,c,e},

{b,c,e},

{b,c,d},

{b,c,d},

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

{a,e},{a,b,c}

{a,e},{a,b,c}

{a,e},{a,b,c}

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

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

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

{a,d,e},{a,b}

{a,d,e},{a,b}

{a,c},

{a,c},

{a,c},

{a,c,e},{a,b}

{a,c,e},{a,b}

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

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

{a,c,d,e},

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

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

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

{a,b,e},

{a,b,e},

{a,b,d},

{a,b,d},

{a,b,d,e},

{a,b,c},

{a,b,c},

{a,b,c,e},

{a,b,c,d},

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

P = **{{}, {a}, {b}, {c}, {d}, {e}, {a,b}, {a,c}, {a,d}, {a,e}, {b,c}, {b,d}, {b,e}, {c,d}, {c,e}, {d,e}, {a,b,c}, {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {a,d,e}, {b,c,d}, {b,c,e}, {b,d,e}, {c,d,e}, {a,b,c,d}, {a,b,c,e}, {a,b,d,e}, {a,c,d,e}, {b,c,d,e}, {a,b,c,d,e}}**

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.

This calculator has 1 input.

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

For more math formulas, check out our Formula Dossier

- 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

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