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} |