Page 81 - ICSE Math 7

6 Set Theory

Key Concepts

• Set • Subsets
• Notation and Representation of Sets • Universal Set
• Types of Sets • Operations on Sets
• Cardinal Number • Cardinal Properties of Sets

The set theory helps us to classify and separate objects, people and ideas according to their properties.
For example, we can classify the students of a class according to various traits such as their weight,
height, and first letter of their names.
In this chapter, we will learn about sets and their representation in roster form and set-builder form. We
will also learn about different types of sets, cardinal number of sets, universal set, subsets, operations
on sets and cardinal properties of sets.

Set
A set is a collection of well-defined and distinct objects and every object of a set is known as its
element or member. By ‘well-defined’ we mean that it should be clear (without any doubt) whether
an object belongs to the set or not. For example,
(a) ‘ The collection of positive numbers less than 11’ is a set because from the given numbers, we can
always find out whether a number belongs to the set or not.
(b) ‘The collection of intelligent girls in your class’ is not a set because in this case there is no rule
by which we can find out whether a girl is intelligent or not.
By ‘distinct’ we mean that no two members of the set will be same. For example, consider the set of
letters in the word GOOD. We observe that O occur twice. So, the collection, i.e., G, O, O, D is not
a set. However, if we take O only once, then the collection of letters, i.e., G, O, D will form a set.

Notation
A set is denoted by capital letters such as A, B, X, Y, etc. and its elements are denoted by small letters
such as a, b, x, y, etc. The symbol ‘∈’ (read as epsilon) stands for ‘belongs to’ or ‘is an element of’
and the symbol ‘∉’ stands for ‘does not belong to’ or ‘is not an element of’. For example, if x is a
member of set X and y is not a member of set X, then we write x ∈ X, y ∉ X.
Example 1: Which of the following collections are sets?
(a) First six months of a year (b) All the toppers of your class

(c) All the negative numbers (d) Four good cricket players of India
Solution: (a) The given collection is a set because the collection contains well-defined and distinct
objects, namely, January, February, March, April, May, June.

(b) The given collection is not a set because there is no rule to decide toppers of a
class and hence the elements (top students) are not well-defined.

(c) The given collection is a set as it contains all the negative numbers.
(d) The given collection is not a set because there is no rule to decide the performance of
cricket players and hence people may differ in their choice in selecting good players.

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