## Power set

Power set is nothing but the set which contains all possible sub sets of a particular set.

Let us consider a set which has been named as set A. There are “n” numbers of elements in the set A. That is “n” is the cardinal-number of the set “A”. Now power-set of A is also a set which is having all the possible sub sets of A and that particular set is denoted as P(A). Since there are “n” umber of elements in the set A, we will have 2^{n} elements in the power-set of A.

That is n[ P(A) ] = 2 ^{n}

In other words, power-set of any set A is set of all subsets of A, including itself and empty set.

Now, let us come to know about “Cardinal-number of a set” in more detail.

**Cardinal-number of a set:**
The cardinal-number of a set is the number of all elements in the set. If there are “n” numbers of elements in the set A, then the cardinal-number of the set A is “n”. In symbolic representation, we have n (A) = 7. The cardinal-number of a set A is denoted by n (A).

** For example:**
In the set A={a,b,c}, the number of elements in A is 3. So, n(A)=3.

To have better understanding about cardinal number-of a set, we can consider the following examples.

In the set B = {0}, there is one element zero. So n (B) = 1, “0” has to be considered as one element.

In the set C = { }, there are no elements. So n (c) = 0.

**Write power-set of 1.A={a,b} 2.B={1,2,3} and also write the number of power sets.**

**Solution:**

1. The proper subsets of A are {a} and {b}.

So,**P(A)={{},{a}, {b},{a,b}}.**

n(P(A))=2^{2}2=2x2=4.

2. The proper subsets of B are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}.

So the **P(B) = {(),{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}**.

.

**n(P(B))=2**^{3}= 2x2x2=8.

.

**Note:**

*Care should be taken to include empty set ({ }) and the set itself in the power-set along with the proper subsets.*

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Related Topics

**
**- Set Theory
- Representation of Set
- types of set
- Disjoint sets
- Operations on Sets
- Laws on set operations
- Venn diagrams
- Set word problems
- Relations and functions

### Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.They are:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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