Different Types of Sets

(i) Empty (Void/Null) Set

A set which has no element, is called an empty set. It is denoted by f or { }.

e.g., A = Set of all odd numbers divisible by 2

and B = {x : x $\in$ N and 5 < x < 6}

Such sets which have atleast one element, are called non-void set.

Note If $\phi$ represents a null set, then $\phi$ is never written with in braces i.e., {$\phi$} is not the null set.

(ii) Singleton Set

A set which have only one element, is called a singleton set.

e.g., A = {x : x $\in$ N and 3 < x < 5} and B = {5}

(iii) Finite and Infinite Sets

A set in which the process of counting of elements surely comes to an end, is called a finite set. In other words ‘A set having finite number of elements is called a finite set’. Otherwise it is called infinite set i.e., if the process of counting of elements does not come to an end in a set, then set is called an infinite set.

e.g., A = {x : x $\in$ N and x < 5}

B = Set of all points on a plane In above two sets A and B, set A is finite while set B is infinite. Since, in a plane any number of points are possible.

(iv) Equivalent Sets

Two finite sets A B and are said to be equivalent, if they have the same number of elements.

e.g., If A = {1 ,2 ,3} and B = {3 ,7 ,9}

Number of elements in A = 3 and number of elements in B = 3

so, A and B are equivalent sets.

(v) Equal Sets

If A and B are two non-empty sets and each element of set A is an element of set Band each element of set Bis an element of set A, then sets A and B are called equal sets. Symbolically, if

x $\in$ A $\Rightarrow$ x $\in $ B and x $\in$ B $\Rightarrow$ x $\in $ A

e.g., A = {1, 2, 3} and B = {x : x $\in$ N, x $\leqslant$ 3}

Here, each element of A is an element of B, also each element of B is an element of A, then both sets are called equal sets.

Note: Equal sets are equivalent sets while its converse need not to be true.

(vi) Subset and Superset

Let A B and be two non-empty sets. If each element of set A is an element of set B, then set Ais known as subset of set B. If set A is a subset of set B, then set B is called the superset of A.

Also, if A is a subset of B, then it is denoted as A B Í and read as ‘A is a subset of B’.

Thus, if x $\in$ A $\Rightarrow$ x $\in $ B,
then A $\subseteq$ B

If x $\in$ A $\Rightarrow$ x $\notin $ B,
then then A $\nsubseteq$ B
and read as ‘A is not a subset of B.’

e.g., If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}

Here, each element of A is an element of B. Thus, A $\subseteq$ B i.e., A is a subset of B and B is a superset of A.

Note

Null set is a subset of each set.
Each set is a subset of itself.
If Ahas n elements, then number of subsets of set Ais $2^n$ .

(vii) Proper Subset

If each element of A is in set B but set B has atleast one element which is not in A, then set A is known as proper subset of set B. If A is a proper subset of B, then it is written as ‘A $\subset$ B ’ and read as A is a proper subset of B.

e.g., If N= {1, 2, 3, 4, …} and
I = {…, –3, –2, –1, 0, 1, 2, 3, …} then
N $\subset$ I

Note : If Ahas n elements, then number of proper subsets is $2^n - 1$

(viii) Comparability of Sets

Two sets A B and are said to be comparable, if either A $\subset$ B or B $\subset$ A or A = B, otherwise, A B and are said to be incomparable.

e.g., Suppose A B = = {1, 2, 3}, {1, 2, 4, 6} and C = {1, 2, 4}
Since, A ⊅ B or B ⊅ A or A $\neq$ B

So, A and B are incomparable.
But C $\subset$ B
B and C are comparable sets.

(ix) Universal Set

If there are some sets under consideration, then there happens to be a set which is a superset of each one of the given sets. Such a set is known as the universal set and it is denoted by S or U.

This set can be chosen arbitrarily for any discussion of given sets but after choosing it is fixed.

e.g., Suppose A = {1, 2, 3}, B = {3, 4, 5} and C = {7, 8, 9}
so, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} is universal set for all three sets.

Sample Problem 1. [NCERT]
Consider the following sets
A = The set of lines which are parallel to the X-axis.
B = The set of letters in the English alphabet.
and C = The set of animals living on the earth.
Which of these is finite or infinite set?
(a) Finite set A → B, , Infinite set → C
(b) Finite set B → C, , Infinite set → A
(c) Finite set A → C, , Infinite set → B
(d) None of the above

Answer: (b)
A = Infinite lines can be drawn parallel to X-axis
B = There are finite 26 English alphabets
C = There are finite number of animals living on earth

Sample Problem 2
Two finite sets have m and n elements, respectively. The total number of subsets of the first set is 56 more than the total number of subsets of second set. What are the values of m and n, respectively?
(a) 7, 6 (b) 6, 3 (c) 5, 1 (d) 8, 7

Answer: (b) Since, total possible subsets of sets A and B are $2^m$ and $2^n$, respectively. According to given condition,

$2^m - 2^n=56$
$2^n(2^{m-n}-1)=2^3 \times (2^3-1)$

On comparing both sides, we get
$2^n$ = $2^3$ and $2^{m-n}=2^3$

n = 3 and m - n = 3
m = 6 and n = 3

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JEE-IIT-NCERT Physics & Math

Different Types of Sets

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