Set

Set

A set is a well-defined collection of objects.

Here “well-defined collection of objects” means that given a specific object it must be possible for us to decide whether the object is an element of the given collection or not.

The objects of a set are called its members or elements.

For example,

1.        The collection of all books in a District Central Library.

2.        The collection of all colours in a rainbow.

3.        The collection of prime numbers.

We see that in the adjacent box, statements (1), (2), and (4) are well defined and therefore they are sets. Whereas (3) and (5) are not well defined because the words good and beautiful are difficult to agree on. I might consider a student to be good and you may not. I might consider Malligai is beautiful but you may not. So we will consider only those collections to be sets where there is no such ambiguity.

Therefore (3) and (5) are not sets.

Both these conditions are natural. The collection 1,2,3,4,5,6,7,8, … as well as the collection 1, 3, 2, 4, 5, 7, 6, 8, … are the same though listed in different order. Since it is necessary to know whether an object is an element in the set or not, we do not want to list that element many times.

Notation

A set is usually denoted by capital letters of the English Alphabets A, B, P, Q, X, Y, etc.

The elements of a set are denoted by small letters of the English alphabets a, b, p, q, x, y, etc.

The elements of a set is written within curly brackets “{ }”

If x is an element of a set A or x belongs to A, we write x ∈ A.

If x is not an element of a set A or x does not belongs to A, we write x ∉ A.

For example,

Consider the set A = {2,3,5,7} then

2 is an element of A; we write 2∈A

is an element of A; we write 5∈A

is not an element of A; we write 6∉A

Example 1.1

Consider the set A = {Ashwin, Muralivijay, Vijay Shankar, Badrinath }.

Fill in the blanks with the appropriate symbol ∈ or ∉.

(i) Muralivijay ____ A. (ii) Ashwin ______ A. (iii) Badrinath ______A.

(iv) Ganguly _____ A. (v) Tendulkar _____ A.

Solution

(i) Muralivijay ∈ A. (ii) Ashwin ∈ A (iii) Badrinath ∈ A (iv) Ganguly ∉ A. (v) Tendulkar ∉ A.