Power Set

Introduction

Relations, Functions, and Power set is the topic of Power Set. It is one of the most important topics for the examination. This chapter will deal with the different properties of power sets and some questions based on them.  In simple terms, A collection of all subsets of set A is called the set of the set A for a particular set A. A power set of set A is denoted by P(A) or 2A. 

So, go through the complete notes of this chapter to get full marks in the question of this chapter in IIT JEE Examinations. 

Power Set

A collection of all subsets of set A is called the power set of the set A for a particular set A. A power set of set A is denoted by P(A) or 2A

P(A)= {x/x ∈ A}

Lets us take an example:-

P(o)={o}

A= {a}, P(A)={o,{a}}

B= {a,b}, P(B)= {o,{a},{b},{a,b}}

C={a,b,c}, P(C)={o,{a}, {b}, {c},{a,b},{b,c},{a,c}{a,b,c}}

This is the reason why the power set of A is sometimes represented by 2A.

In Set A, we have only 1 element.

Therefore, P(A) has 2 =2 element

In Set B, we have 2 components.

Therefore, P(B) has a square of 2=4.

Therefore, if we have n elements in Set A, P(A) must have 2n elements.

What do you mean by Cardinality of power set?

The phrase cardinality means the number of cardinal members in a set. Cardinality can be finite (a non-negative integer) or infinite, and it is denoted as n(A) as the cardinal number of A.

For example: we have a set,

A={a,e,i,o,u}

n(a)=5

B-{letter of word “floor”}

n(a)={f,l,o,r}

While calculating or finding the cardinal number of the set, we omit the repetition of the elements. So, in this case, the cardinal number of B, i.e. n(B), will be equal to 4.

What is the Cardinal number 0?

Zero (0) is not a cardinal number, since it means nothing.

What are the Properties of a Power Set?

The power set is the set that has a list of all the subsets of a given set. The power set that has represented by P(A) with ‘n’ objects or elements has the following properties:

• Power set defines the empty set of definite elements 
• The power set of a blank set has only 1 element
• In Sets, a Finite number of elements is finite. Let’s take an example, where set P = {e,f,g}, here, the power of sets is countable
• The power set of an infinite set has an infinite number of subsets
• It can be much greater than the real set
• The number of objects or elements consist in a power set of P is 2n, here n denotes number of objects or elements present  in set A
• For the natural numbers set, we could do one-to-one mapping of the final or resultant set, P(A), with the natural numbers
• P(A)  of sets, when operated with a union of sets, the intersection of sets & complement of sets, represents the concept of Boolean Algebra

Describe Power Set of Empty Set.

The empty set had 0 or void elements. So, the power set of a void or empty set { }, can be described as;

• A set consist void set
• It consists of null or zero elements
• The void set or empty set is the subset

If the number of elements in a set is ‘n’, then there will be 2n elements in the power set. The empty set is a set with no elements. It is denoted by { } or the symbol Ø. This implies, { } is a subset of every set. An empty set does not consist of any element. Therefore, the void or empty set’s power set is an empty set only.

Above, we have learned that a void or empty set does not contain any elements, this implies the power set of an empty set will consist of 20 elements. Here, the void or empty set’s power set is an empty set with one object, i.e., 20 = 1. So, P(E) = {}.

What is a Recursive Algorithm?

A recursive algorithm is called recursive if it solves a problem by decreasing it to an instance of a similar problem with a smaller input. 

In other words, an algorithm is said to be recursive if the same algorithm is invoked in the body.

Give a recursive algorithm for computing the greatest common divisor of the two non-negative integers a and b with a,b.

Procedure gcd (a,b: non negative integers, a<b)

If a=0 then return b

Else return gcd (b mod a,a)

{output is gcd (a,b)}

Let’s use a recursive algorithm for finding, where n is a non-negative integer.

Procedure factorial (n: non-negative integer)

If n=o,invoked return 1

Else return n: factorial (n-1)

{output is n!)

Let’s use a recursive algorithm for finding an, where a is a non-zero real number and n is a non-negative integer.

Procedure power i.e. an (a is non zero real number, n is a non-negative integer)

If n=0, then return 1

Else return a,=. Power (a, n-1)

{output is an}

Types of Recursion Algorithms

There are mainly two types of Recursion Algorithms:

Direct Recursion Algorithms: Algorithm A is said to ve direct recursive if it calls itself.

Indirect Recursion Algorithms: The algorithm is called Indirect Recursion if it calls another algorithm which in turn is called A.

Every recursive algorithm has two elements, i.e., Base case where the statement solves the problem, and the recursive function must have a base case. Another is a General case where each call reduces the size of the problem. The rest of the function is known as the general case. 

Conclusion

In the above, we have studied the properties, functions, and types of Relations.  In simple terms, A collection of all subsets of set A is called the power set of the set A for a particular set A. A power set of set A is denoted by P(A) or 2A. This chapter is the core foundation for another chapter like sets, complex numbers and functions. 

The relation is also important from the examination point of view and in the 2019 IIT JEE examination there are around 3-4 questions asked on this topic. So, go thoroughly through the notes to secure all marks from this chapter in the next examination.