Page 103 - ICSE Math 6

7 Set Theory

Key Concepts

• Idea of Sets • Types of Sets: Finite, Infinite and Empty
• Representation of Sets • Cardinality of a Set

Set

In our daily life, we come across the word ‘collection’ in many contexts such as a collection of tickets,
a collection of coins, etc. A collection of well-defined objects is known as a set and every object
forming a set is called a member or an element of the set. Well-defined means that it should be clear
whether an object belongs to the set or not. For example,
(a) ‘All the students of a class’ is a set because every student will be included in the set and hence
the collection is well-defined.
(b) ‘All the tall girls of a basketball team’ is not a set because the minimum height of a tall girl is not
decided and hence the collection is not well-defined.
(c) ‘Collection of all the prime numbers’ is a set because every prime number will be included in the
set and hence the collection is well-defined.

Now, consider a collection of all the letters in the word MATHEMATICS. We observe that M, A and
T occur twice. So, this collection is not a set. However, it can be expressed as a set by taking the
letters M, A, T, H, E, I, C, S only once. Thus, if a collection has the same object occurring twice, then
consider each object only once to express the collection as a set.

Notation
A set is denoted by a capital letter such as A, B, C, etc., and the members of a set are denoted by small
letters such as x, y, z, etc.
The Greek symbol ∈ (read as epsilon) which means ‘belongs to’ or ‘is an element of’ is used for
representing elements of a set. For example, if x is a member of set A, then we express it by writing
x ∈ A (read as x belongs to A). Also, if y is not a member of A, then we write y ∉ A (read as y does
not belong to A).

Example 1: Which of the following collections are sets? Give reasons.
(a) A collection of beautiful flowers.

(b) All the boys of your class whose height is more than your height.
(c) All difficult problems of your textbook.
(d) The collection of distinct letters of the word MUMBAI.

Solution: (a) The given collection is not a set because the objects (i.e., beautiful flowers) are not
well-defined as the word ‘beautiful’ is a relative term.
(b) The given collection is a set because it can be easily checked whether the height
of a boy is more than your height or not and hence the elements are well-defined.

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