Sets:Types of Sets, Subset, Superset, Power Set and Universal Set

The diagram below shows a fruit basket A which contains two more baskets B and C inside it. But can you guess why we are talking about such a situation? 

Mathematically if we consider $A,B \& C$as three sets then we can state that $B \& C$ are subparts of A. In set theory we have special names for these kinds of sets. Let’s study about these special kinds of sets in detail.

Table of Contents

• Types of Sets
• Subset
• Superset
• Power set
• Universal set
• Practice Problems
• FAQ’s

Types of Sets

• Empty Set

A set is said to be a null set or void set or empty set if it contains no element. An empty set is 

represented by $\left\{ \right\}$ or $\varphi$                     

Example: $A=$ set of natural numbers lying between 1 and 2 is an empty set 

• Singleton Set

A set containing only one element is known as a singleton set. 

Example: $P=y\in R:y-\surd 7=0=\surd 7,$, which is a singleton set

• Finite Set

A set is called a finite set if it contains either no element or a finite number of elements.

Example: A = the set of natural numbers less than 10 i.e. $A=\left\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},\mathrm{7,8},9\right\}$

• Infinite Set

A set containing an infinite number of elements is known as an infinite set.

Example: N = the set of all natural numbers i.e. $N=\left\{ \mathrm{1,2},\mathrm{3,4},5\dots \dots \dots ..\right\}$

• Equivalent Sets

Two finite sets A and B are said to be equivalent sets if their order or cardinalities are the same, i.e., $n\left(A\right)=n\left(B\right)$.

For example, $A=\{\mathrm{1,2},\mathrm{3,4}\} \& B=\{a,b,c,d\}$

, hence sets A and B are equivalent.

• Equal Sets

Two sets are said to be equal if the cardinalities as well as the members in both the sets are the same. It is not necessary that the elements are listed in the same sequence in both the sets.

For example, $P=\{a,b,c,d\} \& S=\{b,c,a,d\}$. Here $P \& S$ are equal sets. 

• Disjoint Sets

The two given sets are said to be disjoint , if they do not contain any common elements.

Example: $P=\{\mathrm{1,2},3\} \& S=\{\mathrm{5,6},\mathrm{7,8}\}$. Here $P \& S$ are disjoint sets.

Subset 

Let S and P be two sets. If every element/member of S is an element of P then S is called a subset of P.If a set contains n elements, then number of subsets=2ⁿ

Notation: s ⊆ p is read as “s is a subset of p” or “s is contained in p”.

If p is not a subset of s, we write p ⊈ s.

Example: $A=\left\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},\mathrm{7,8},9\right\}$ and $B=\left\{\mathrm{1,2},\mathrm{3,9}\right\}$ hence B ⊆ A.

Subsets are of the following two types:

$\left(\mathit{i}\right)$Proper Subset 

If A ⊆ B and $A\ne B$ , then A is called the proper subset of B and it can be written as A ⊂ B.

For fig$\left(\mathit{i}\right)$shown above, B ⊂A & C⊂ A

Example: If $A=\left\{\mathrm{2,5},7\right\}$  and $B=\left\{\mathrm{2,5},7\right\}$ then A is not a proper subset of B

But if $A=\left\{\mathrm{2,5}\right\}$ and $B=\left\{\mathrm{2,5},7\right\}$ then A is a proper subset of B i.e. A ⊂ B

Note: Every set is not a proper subset of itself.

$\left(ii\right)$ Improper Subset 

An improper subset is a subset containing every element of the original set.Every set is an improper subset of itself.

Example: If $P=\left\{k,l,m,y,o\right\}$, then the improper subset of P is any set equal to Set P, i.e., $B=\left\{m,o,y,l,k\right\}$ is the improper subset of P 

Note : Empty set is an improper subset to itself and a proper subset to any other set.

Superset

Set P is said to be the superset of S if all the elements of set P are also the elements of set P. It is represented as P ⊃  S.

 Example: If $A=1,2,3,4$ and $B=\left\{\mathrm{1,3},4\right\}$, then A is the superset of B.

Power Set

Let 𝐴 be any set. Then the collection or family of all subsets of is called Power Set of 𝐴 and it is denoted by $P\left(A\right)$. 

Example : If $A=\left\{\mathrm{1,2},3\right\}$, then $P\left(A\right)=\left\{\varphi ,\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{\mathrm{1,2}\right\},\left\{\mathrm{2,3}\right\},\left\{\mathrm{3,1}\right\},\left\{\mathrm{1,2},3\right\}\right\}$

Mathematically $P\left(A\right)=\left\{S:S \subseteq A\right\}$ 

Note: Let B be a finite set containing 𝑚 elements i.e. 𝑛(B)= K, then 

• Number of elements in the power set of B, $n\left(P\left(B\right)\right)=2^k$.
• Number of elements in the power set of power set of B, $n\left(P\left(P\left(B\right)\right)\right)=2^{2^k}$ and so on.
• Number of non-void/non-empty subsets of $B=2^k-1$.
• Number of proper subsets of $B=2^k-1$.
• Number of non-void proper subsets of $B= 2^k-2$.

Universal Set

A set which contains all the elements of the other sets under consideration is known as the universal set for those sets.It is denoted by U

Example: If $A=\left\{\mathrm{1,2},3\right\}$ and $B=\left\{\mathrm{2,3},\mathrm{4,5}\right\}$, the universal set here will be: $U=\left\{\mathrm{1,2},\mathrm{3,4},5\right\}$

Practice Problems of Set

Example : Find Power Set of set $X=\left\{3, 9, 11\right\}$ and the total number of elements.

Answer:

Given, $X=\left\{3, 9, 11\right\}$

Total number of elements of $X=3$.

Total number of elements of $P\left(X\right)=2^n=2^3= 8$.

$P \left(X\right)=\left\{ \left\{\right\}, \left\{3\right\}, \left\{9\right\}, \left\{11\right\}, \left\{\mathrm{3,9}\right\}, \left\{\mathrm{3,11}\right\}, \left\{\mathrm{9,11}\right\}, \left\{\mathrm{3,9},11\right\}\right\}$

Example : If a set $A=\left\{x : x is a prime number less than 4\right\}$ then find the number of elements in  $P\left(P\left(A\right)\right)$

Answer:

From the given question, $A=\left\{\mathrm{2,3}\right\}$ Power set of A has $2^2=4$ elements i.e., $n\left(P\left(A\right)\right)=4$

So, the number of elements in $P\left(P\left(A\right)\right)=4^2=16$

Example : Let $A=\left\{\mathrm{1,2}\right\}, B=\left\{\mathrm{2,4}\right\}, C=\left\{\mathrm{4,5},6\right\}.$ Which of the following may be considered as the universal set for set $A, B, C ?$

$a\left) \right\{\mathrm{1,6},\mathrm{7,8},9\}$

$b\left) \right\{\mathrm{1,2},\mathrm{3,4}\}$

$c\left) \right\{\mathrm{2,4},\mathrm{5,6}\}$

$d\left) \right\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\}$

Answer:

A Universal set is the set which is a superset of all basic sets of that type.

$\left\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\right\}$ is the set which contains all the elements of set $A, B, C$. So, it's the Universal set of all the sets.

Hence, option (d) is correct.

Example : If the set S in each part is finite, write down $\left|S\right|$.

$\left(i\right)S=\left\{\pi \right\}$

$\left(ii\right)S=\left\{decimal places of \pi \right\}$

$\left(iii\right)S=\left\{whole numbers less than \mathrm{100,000,000}\right\}$

Answer:

$\left(i\right)$ As $S=\pi$ contains 1 element, $\left|S\right|=1$

$\left(ii\right)$ The number $\pi$ itself is rounded up to 3.14 but it can go on forever(non-terminating). Hence the set containing the decimal places is not finite. 

$\left(iii\right)$ The set will contain whole numbers from 1 to $\mathrm{99,999,999}$. Hence, $\left|S\right|$ is $\mathrm{99,999,999}$.

FAQs of Set

Question 1.What are the subsets of complex numbers?

Answer: Real numbers, Rational numbers, Irrational numbers, Integers, Natural numbers, Whole numbers are the subsets of a complex number.

Question 2. How many elements can a set contain?

Answer: A set can contain an infinite number of elements.

Question 3.Do all complex numbers form a set?

Answer: Since complex numbers are well defined, they can form a set.

Question 4.Is there any difference between Superset and Universal Set?

Answer: Yes, A superset contains all the elements of a particular set whereas an Universal set contains all the elements of all the sets under consideration (including the superset)

Related Topics to Types of Sets in Maths

• Sets
• Venn Diagram
• Operation of Sets

NCERT Class 11 Maths Chapters