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Sets: Definition, Elements and Notation of Sets

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,nN}

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(A),|A|, O(A) 

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|t3=t,tR in the roster form.      

Answer:

Here, setD=t|t3=t,tR

Now, setD=t|t3=t,tR

 tt2-1=0

tt+1t-1=0      [a2-b2=a-ba +b]

t=0, 1, -1

Hence, the roster form is  D={-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  Y but a2Y. Also state the cardinality of the obtained set.

Answer:

Here,Y={1,2,3,....10} and aY

Since12, 22,32Y

 a :a  Y and a2Y={4,5,6,7,8,9,10} 

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={x | x  R, 4  x<5}

Example : Let B=2m+5n|m,nN.. 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 : nZ and n2≤4}and B={x : xR and x2-3x+2=0}

Answer:

Step 1 : 

 

Solve for elements of set A

 

A={n : nZ and n24}

Since, n2≤4

The elements are -2,-1,0,1,2

A={-2,-1,0,1,2}

Step 2 : 

 

Solve for elements of set B

 

B={x : xR and x2-3x+2=0}

 

Solving,x2-3x+2=0

x2-2x-x+2=0

⇒(x-2)(x-1)=0

x=1,2

B={1,2}

 

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

NCERT Class 11 Maths Chapters

Sets Relations and Functions Triginometric Functions
Mathematical Induction Numbers and Quadriatic Equations Linear Inequalities
Premutations and Combinations Binomial Theorem Sequence and Series
Straight Lines Conic Sections 3 D Geometry
Limits and Derivatives Mathematical Reasoning Statistics
Probability  
 
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