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Power Set

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Define Power Set with Example

Well! To define a power set, we would say it is simply a set of all the subsets of a set. Feeling confused? Let’s understand the power set with an example. Let’s say:

An ice cream parlor has strawberry, pineapple and butterscotch flavors

What order do you place?

  • Absolutely Nothing: {}

  • Or maybe just strawberry: {strawberry}. Or just {pineapple} or just {butterscotch}

  • Or two together: {strawberry, pineapple} or {pineapple, butterscotch} or {butterscotch, strawberry}

  • Or all three! {pineapple, butterscotch, strawberry}

Collectively, we get the Power Set of {x, y, and z}:

P(S) = {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z} }


Subsets

Subset is a collection of sets or we can say all the different ways we can select the products. The order of the items being selected doesn't matter, even selecting none, or all.

For the set with variables {x, y, and z}:

The empty set {} is a subset of {x, y, z}

These are subsets: {x}, {y} and {z}

These are also subsets: {x, y}, {x, z} and {y, z}

And {x, y, z} is a subset of {x, y, z}


Number of Subsets

It’s very simple! Let’s assume if the original set consists of ‘n’ members, then the Power Set will contain 2n members. This is to say:

{w, x, y, z} has 4 members (w,x,y and z).

Thus, the Power Set must have 24 = 16.


Power Set of a Countable Set

We call a set countable, when its element can be counted. Note that a countable set can be both finite and infinite.

For example, let’s take two sets

S1 = {B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Z and Y} representing all consonants is a countably finite set.

On the other hand, S2 = {1, 2, 3, 4, 5 , 6,……} depicting a set of natural numbers is a countably infinite set.

That said, Power set of countably finite sets is always finite and thus countable.

For example, set S1 representing consonants has 21 elements and its power set will have 221 = 2,097,152 elements. Hence, it is finite and therefore countable.

Power set of countably infinite sets is uncountable.


Power Set of an UnCountable Set

We call a set uncountable, when its element is unable to be counted. Note that an uncountable set is always infinite. Remember that the Power set of an uncountable set is always uncountable.

For example, set S3 depicting all fractional numbers between 1 and 10 is uncountable. Hence, the power set of uncountable sets is uncountable invariably.


Solved Examples

Solve the following question based on the power set.


Example 1: What will be the Cardinality of the Power Set of {0, 1, 2 . . ., 5}?

Solution:

The cardinality of a set is the number of elements that it consists of.

Given that, for a set S with 6 elements, its power set will be

n= 6

Now, the size of power set is 26 = 64


Example 2: In a food joint, we have our all time favorite food: pizza, macaroni, sandwich and burger. Evaluate the number of different ways can we have them?

Solution:

Let's use letters for the foods as: {P, M, S, B}. Selections include:

{} (nothing, you avoid fast food)

{P, M, S, B} (Every food item)

{M, S} (Macaroni and Sandwich are good together) etc

Let’s create a table using binary


SL.NO

PMSB

Subsets

0

0000

{}

1

0001

{m}

2

0010

{s}

3

0011

{m,s}

...

...etc...

...etc..

12

1100

{p,m}

13

1101

{p,m,b}

14

1110

{p,m,s}

15

1111

{p,m,s,b}


And the arranged outcome is as follows:

P = { {}, {p},{m},{s},{b},{m,s}, {s,b}, {b,p}, {p,m}, }, {m,b}, {s,p}, {p,m,b}, {p,m,s}, [m,s,b}, {p,m,s,b}

FAQ (Frequently Asked Questions)

Q1. How Do We Represent a Power Set?

Answer: For a given set N, Power Set P(N) or 2n represents the set having all possible subsets of N as its elements.

Q2. How is a Power Set Useful?

Answer: The Power Set can be quite useful in complicated and unexpected areas such as:

  • To find all factors and even the prime of a number.

  • Test all possible numbers: 0, 1 2, 3, 4, 5, 6, 7, and 8,9,10,100 etc…

  • Combining prime factors

Q3. What are the Properties of a Power Set?

Answer: Following are the properties associated with a power set:

  • Power set of a finite set is always finite.

  • Set ‘S’ is an element of power set of ‘S’ which can be mathematically expressed as S ɛ P(S).

  • Empty Set denoted by ɸ is an element of power set of S which can be mathematically expressed as ɸ ɛ P(S).

  • Empty set denoted by ɸ is subset of power set of S which can be mathematically expressed as ɸ ⊂ P(S).

Q4. How Binary Systems are Used to Represent a Power Set?

Answer: In order to form the Power Set, we need to write down the sequence of binary numbers (using n digits), and then let "1" mean "inserting the matching member into that subset".

So "101" is replaced by {1 for a}, {0 for b} and {1 for c} to have us {a,c}