## 1.2 Set and their Representations

Set: A set is a well-defined collection of objects.

(i) Objects, elements and members of a set are synonymous terms.

(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.

If a is an element of a set A, or “a belongs to A” then the Greek symbol (epsilon) is used to denote the phrase belongs to’.

∈ (epsilon)   =  belongs to

Thus, we write a A         [element a belongs to set A]

If ‘b’ is not an element of a set A, we write -

b A and read b does not belong to A”.

Set representation methods:-

(i) Roster or tabular form

(ii) Set-builder form.

Roster form: - All the elements of a set are listed, the elements are being separated by commas (,) and are enclosed within braces { }.

For example,

The set of all even positive integers less than 7 = {2, 4, 6}

The set of all natural numbers which divide 42 = {1, 2, 3, 6, 7, 14, 21, 42}

The set of all vowels in the English alphabet = {a, e, i, o, u}

The set of odd natural numbers = {1, 3, 5, . . .}

Note:- The dots tell us that the list of odd numbers continue indefinitely.

Note:- In roster form, the order in which the elements are listed is immaterial.

Note:- while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.

Set-builder form: - All the elements of a set possess a single common property which is not possessed by any element outside the set.

For example,

In the set {a, e, i, o, u}, all the elements possess a common property (each of them is a vowel) and no other letter possess this property.

Denoting this set by V, we write V = {x : x is a vowel in English alphabet}    (set-builder form)

A = {x: x is a natural number and 3 < x < 10}         (set-builder form)

Here “A is 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.

 Roster Form (RF) Set Builder Form (SBF) A = {1, 2, 3, 6, 7, 14, 21, 42} A= {x : x is a natural number which divides 42} V = {a, e, i, o, u} V= {y : y is a vowel in the English alphabet} B = {1, 3, 5, . . .} B = {z : z is an odd natural number}