Sets: Definition, Elements and Notation of Sets

In day-to-day life, we often discuss the collection of objects of a specific kind, like, collection of coins, collection of books, players of a cricket team, etc.

So, can we say that this is a collection of good cricket players or a set of good cricket players? The answer will vary from person to person. Hence this is not a well defined collection.

Similarly, we come across some collections in mathematics as well. For example: collection of irrational numbers, lines, composite numbers, etc. Let’s understand when we can call a collection of things a set.

Table of Contents:

• Definition of Set
• Elements of a set
• Notation of Sets
• Cardinality of a Set
• Practice Problems
• FAQs

Definition of Set 

A set is a  well-defined collection of elements or objects. A set is always denoted by a capital letter. Some examples of sets:

• R: Set of all real numbers
• N:  Set of all natural numbers
• The collection of all states of India

Note: A collection of good students is not a set as the term “good” is vague i.e. not well defined.

Elements of a Set 

The objects in a set are called its members/elements. Elements can be present in any order and in general, we won't repeat elements in a set.

If a is an element of set A then we write a∈  A and is read as “a belongs to A”.

If a is not an element of set ‘A’ then we write a ∉  A and is read as “a does not belong to A”.

Notation of Sets

A set is represented within curly braces { }. There are two common ways of representing the set:

• Roster or Tabular Notation

In Roster form, all the elements of a set are listed, separated by commas and enclosed within braces.

Example: A= the set of even natural numbers will be represented in Roster/Tabular form as  A = {2,4,6,8,....}

• Set Builder Notation

In the set-builder form, property or properties which are satisfied by all the members of the set are listed. We write, {x:x satisties property P}, which is read as “ set of all those such that (represented by | or :) , each x has property P 

Example: A= The set of even natural numbers will be represented in set builder form as  $A=\{x:x=2n,n\in N\}$

Cardinality of a Set

The number of elements present in a given set is known as the cardinality or order of that set.The order/cardinality can be finite or infinite depending on the number of elements present in the set. The cardinality of a given set A is denoted by $n\left(A\right),\left|A\right|, O\left(A\right)$ 

Example: Let a set A= {1,2,3,4} then cardinality or order of set A is n(A) = 4.

Practice Problems of Sets

Example : Write the set $setD=t|t^3=t,t\in R$ in the roster form.      

Answer:

Here, $setD=t|t^3=t,t\in R$

Now, $setD=t|t^3=t,t\in R$

$\Rightarrow t\left(t^2-1\right)=0$

$\Rightarrow t\left(t+1\right)\left(t-1\right)=0$ $[⸪a^2-b^2=\left(a-b\right)\left(a +b\right)]$

$\Rightarrow t=0, 1, -1$

Hence, the roster form is  $D=\{-\mathrm{1,0},1\}$

Example : If Y={1,2,3,....10} and a represents any element of Y, write the set containing all the elements satisfying the condition, $a \in Y but a^2\notin Y$. Also state the cardinality of the obtained set.

Answer:

Here,Y={1,2,3,....10} and a ∈ Y

Since$1^2, 2^2,3^2\in Y$

$\therefore \left\{a :a \in Y and a^2\notin Y\right\}=\{\mathrm{4,5},\mathrm{6,7},\mathrm{8,9},10\}$ 

Hence, the required set is {4,5,6,7,8,9,10} & since the required set contains seven elements,

Therefore, n(A) = 7

Example : Let A be the set of all real numbers between 4 and 5, including 4 but not 5. Describe A in set builder notation, using the variable name as x.

Answer:

In the set builder form, we could write $A=\left\{x \right| x \in R, 4 \le x<5\}$

Example : Let $B=2m+5n|m,n\in N.$. Is 10∈ B? Is 13 ∈ B? Explain.

Answer:

For different values of mand n, we can see B={7, 9, 11, 12,13, 14, 15, 16 ...}. From this list we can say that 10 ∉ B, as no natural numbers mand n, satisfies  2m+5n=10. However,

13 ∈ B, as for m=4and n=1, 2m+5n= 13

Example : Describe sets A and Bin roster form, where

A={n : n∈Z and n2≤4}and B={x : x∈R and x2-3x+2=0}

Answer:

FAQs of Sets

Question 1. Can the cardinality of a set be equal to zero?

Answer: If the set is empty i.e. it contains no elements then the cardinality of that set is zero.

Question 2. Can we represent sets using diagrams?

Answer: Yes, sets can be represented using Venn Diagrams(Possible hyperlink).

Question 3. Can small letters be used to represent a set?

Answer: A set is always represented by a capital letter while its elements are written in small letters

Related Topics to Sets in Maths

• Types of Sets
• Venn Diagram
• Operation of Sets

NCERT Class 11 Maths Chapters