Mathematics Sets and their Representations, - Basics and Definitions
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### Topics Covered

star Sets
star Representation of a Set

### Sets

color{purple}ul(✓✓) color{purple} mathbf(" DEFINITION ALERT")

• A color{blue} ul(mathtt ("Set")) is a WELL-Defined collection of objects. ( called elements of sets )

color{green} ✍️ color{green} mathbf("KEY CONCEPT")

"Meaning of Well Defined - "
A set is well-defined if
• There is no ambiguity as to whether or not an object belongs to it,
• i.e we can always tell what is and what is not a member of the set.

"-Examples of Well Defined collection "
• collection of all vowels
• collection of integers

"-Examples of NOT Well Defined collection "
• collection of 5 most intelligent persons in Delhi
• collection of 5 most cute babies in JanakPuri

\color{green} ✍️ \color{green} \mathbf(KEY \ POINT)
star If color{green} (a \ \ "is an element of a set A"), we say that "“ a belongs to A”."

color{blue}( "The Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. ")
Thus, we write a ∈ A.

• If b is not an element of a set A, we write b ∉ A and read "“b does not belong to to A”."

\color{green} (✍️ "Some Examples of set")
N : the set of all natural numbers

Z : the set of all integers

Q : the set of all rational numbers

R : the set of real numbers

Z^+ : the set of positive integers

Q^+ : the set of positive rational numbers, and

R^+ : the set of positive real numbers.

color{green} ✍️ color{green} mathbf("KEY POINTS")
(i) If same element comes twice or more time in set, then it is consider only once at final set.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.
Q 1954323254

Which of the following are sets? Justify your answer.

(i) The collection of all the months of a year beginning with the letter J.

(ii) The collection of ten most talented writers of India.

(iii) A team of eleven best-cricket batsmen of the world.

(iv) The collection of all boys in your class.

(v) The collection of all natural numbers less than 100.

(vi) A collection of novels written by the writer Munshi Prem Chand.

(vii) The collection of all even integers.

(viii) The collection of questions in this chapter.

(ix) A collection of most dangerous animals of the world.
Class 11 Exercise 1.1 Q.No. 1
Solution:

(i) It is a set. The set of months in a year beginning with J is text({ January , June , July }).

(ii) The collection often most talented writers of India is not a set, because the term 'most talented' is not well defined.

(iii) A team of eleven best-cricket batsmen of the world is not a set, because the term 'best-cricket batsman' is not well defined.

(iv) The collection of all boys in your class is a set, because the boys in your class are well defined.

(v) The collection of all natural numbers less than 100 is a set, because natural numbers less than 100 are {1, 2, 3, ... , 99}.

(vi) The collection of novels written by the writer Munshi Prem Chand is a set, because this collection is well-defined.

(vii) The collection of all even integers is a set, because even integers are : { ... , -4, -2, 0, 2, 4, .. }.

(viii) The collection of questions in this chapter is a set, because the questions in this chapter are well-defined.

(ix) A collection of most dangerous animals of the world is not a set, because the term 'most dangerous animal' is not well-defined.

### Representation of a Set :

There are two methods of representing a set :

(i) Roster or tabular form
(ii) Set-builder form

\color{green} {(i)" Roster or tabular form :"}
\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"  In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

E.g. the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}.

Some more examples of representing a set in roster form are given below :

(a) The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}.

(b) The set of all vowels in the English alphabet is {a, e, i, o, u}.

(c) The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots tell us that the list of odd numbers continue indefinitely.

color{green} {"Note"} It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.

For example, the set of letters forming the word "‘SCHOOL’" is { S, C, H, O, L} or {H, O, L, C, S}. Here, the order of listing elements has no relevance.

\color{green} {"(ii) Set-builder form :"}
\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"  In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

\color{green} ✍️ \color{green} \mathbf( Method :) - We describe the element of the set by using a symbol x, which is followed by a colon "“ : ”."
- After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.

E.g. In the set {a, e, i, o, u}, is a vowel in the English alphabet. Denoting this set by V, we can write as V = {x : x \ \ "is a vowel in English alphabet}" in set builder form.

E.g. A = {x : x \ \ "is a natural number and" \ \ 3 < x < 10} is read as “the set of all x such that x is a natural number and x lies between 3 and 10.
Hence, the numbers 4, 5, 6, 7, 8 and 9 are the elements of the set A.

If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set-builder form as follows:
A= {x : x " is a natural number which divides" 42}

B= {y : y "is a vowel in the English alphabet" }

C= {z : z "is an odd natural number" }

Q 3087167087

Write the set {x : x \ \ "is a positive integer and" \ \x^2 < 40} in the roster form.

Solution:

The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form
is {1, 2, 3, 4, 5, 6}.
Q 3037267182

Write the set {1/2,2/3,3/4,4/5,5/6,6/7} in the set-builder form.

Solution:

We see that each member in the given set has the numerator one less than
the denominator. Also, the numerator begin from 1 and do not exceed 6. Hence, in the
set-builder form the given set is

{ x: x = (n)/(n+1), "where is a natural number and" 1 leq n leq 6}