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# Power Set|Definition & Meaning

## Definition

The power set of a set S is **another set** that contains all the **possible subsets** of the **original set**. It is denoted as 2$^S$ or P(S). For example, let F = {guava, kiwi, berry}. Then, P(F) = **{ **{}, {guava}, {kiwi}, {berry}, {guava, kiwi}, {guava, berry}, {kiwi, berry}, {guava, kiwi, berry} **}**. The power set contains the **full set** and the **empty/null** set, which are valid subsets by definition.

The **cluster** of all **subsets** of a full set S is a collection of sets. Therefore, the **power** set of a provided set is always **full.** This set is stated to be the power set of S and is **indicated** by P(S). If S **possesses** N components, then P(S) has $2^n$ **subsets.**

## What Is a Power Set?

A power set is **expressed** as the set or set of each and every subset for a **piece** of provided set, as well as the **vacant** set, which is represented by {}, or $\phi$. A set that contains ‘n’ **elements** has 2n possible subsets in full.

For instance, let Set A = {1,2,3}. Hence, the **whole** numeral of elements in the set is 3. **Consequently,** the power set **contains** $2^3$ elements. Let us find the power set of A.

A = {1, 2, 3}

**Subsets** of Set A = {}, {3}, {2}, {1}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

**Power** set P(A) = **{** {}, {3}, {2}, {1}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}** }**

## Visual Example of a Power Set

Let’s glance at a **pictorial** to witness just how **multiple** subsets you can obtain out of a set. I am intending to use **ice cream** for this sample: strawberry, chocolate, and vanilla. Noted **algebraically** utilizing abbreviations, the set would seem like this: {S, C, V}. So, how **numerous** subsets can we gain out of this three-element set?

The query can actually be presented as “In how many **distinguishable** methods can these three articles be organized”?

Foremost, you own a **subset** of zero, or { }. Remember that a **null** set is a subset of every set.

At the moment, you keep a **subset** comprising each flavor by itself: {S}, {C}, and {V}. The following set of **subsets** would be the distinct **union** of two options each, such as {S, C}, {S, V}, and {C, V}. The last subset is the **subset** that comprises all the elements: {S, C, V}. These eight **alternatives** are each and every **probable** subset for the set {S, C, V}.

**Concurrently,** each and every one of these subsets makes a power set. A **power** set is one that includes all of the probable subsets of a given set. The **power** set is mentioned like this: P(S) =. So, for our ice cream illustration, the power set memo would be:

P(S,C,V) = {{ }, {S}, {C}, {V}, {S, C}, {S, V}, {C, V}, {S, C, V}}

Did you see that the **sets** of all of the elements and none of the **elements** are subsets, as well as every mixture of the elements? That is **necessary** to recall.

## Numerical Example of a Power Set

Why would you ever require to **utilize** power sets in algebra (or in **actual** life, for that matter)?

Also, power sets can be **utilized** to specify all the various ways things can be placed **concurrently.** Perhaps you ought to sort out the various **probable** groups for a set of learners, or possibly you want to **calculate** how to identify a **group** of objects in a store **window;** you can utilize power sets to determine all the **diverse** ways you could group your objects concurrently.

While **another** usage for them is deciding each and every factor of a number. Power sets are an **outstanding** means to discover every single element for any digit possible.

Okay, so can you describe to me all of the **elements/factors** behind the number 130? (Remember that a factor is a number that can be **multiplied** by another digit to result in the presented product.)

To **decide** all of the factors for 130, we first begin by pinpointing the prime factors, which are those **factors** that are only divisible by themselves, and 1.

The **set of prime factors** for 130 is (S) = {2, 5, 13}

Now, we **develop** a table to explain the power set of this set of prime **factors.**

The **foremost** row in the table is the blank set (empty set), and a factor of 1 is **allocated** because 1 is the multiplicative identity.

The following **three** rows are the **subsets** having a single element; the real element will be the **resultant** factor. The successive three rows are **subsets** with only 2 elements. The factor enumerated is the by-product of the two elements in the subset.

**Ultimately,** you have the subset, which comprises all the elements from the **actual** set, and the factor is, certainly, the allocated **number** from the sample.

## The Cardinality of the Power Set

**Cardinality** depicts the full **number** of elements existing in a set. In the matter of a power set, the **cardinality** will be the list of the **number** of subsets of a set. The numeral of elements of a power set is **noted** as |P (A)|, where A is any set. If A has ānā elements, then the procedure to **discover** the number of subsets of a set in a power set is **presented** by:

|P(A)| = 2n

For instance, set A = {a, b, c}

n = number of elements of A = 3

So, the numeral of subsets in a power set of A will be:

Subsets of A = {}, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}

P|A| = 23 = 8

Therefore, P(A) is {{}, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}

## Power Set Properties

A power set is one that **holds** a checklist of all the subsets of a **provided** set. The power set, which is indicated by P(A) with ‘n’ elements, has the given qualities:

- The
**whole**number of elements of a set is 2^{n}. - A
**null**set is an actual element of a power set. - The power set of a null set has
**only one**element, and that is the**null set itself**. - For a set with a
**limited**number of elements, the power set is**finite**. For instance, the power sets are**enumerable**if set X = {b,c,d}. - The power set of an unlimited set carries an infinite number of subsets. For
**illustration,**if Set X holds each and every multiple of 10 starting from 10, then we can state that Set X has an endless**number**of elements. Although there is an**indefinite**number of elements, a power set yet exists for set X. In this situation, it has a**limitless**number of subsets. - The power set
**subsists**for both finite and infinite sets.

## Solving an Example Problem of a Power Set

Q.1: Discover the power of the below set and also encounter the entire number of elements.

**Solution**

Given that X = {2, 3, 8}

The whole numeral of elements in the power set is 2$^n$

The **entire** number of elements in this situation is n = 3.

So, 2$^3$ = 8, which marks eight elements of the power set of X exist.

**Consequently:**

P(X) = {{}, {2}, {3}, {8}, {2, 3}, {3, 8}, {2, 8}, {2, 3, 8}}

*All images/mathematical drawings were created with GeoGebra.*