ICSE Class 6 Maths Chapter 26 Idea of a Set

Unit - 7: Set Theory

Chapter 26: Idea Of A Set

26.1 Introduction

In our day-to-day life we often speak or hear about different types of collections.

Such as:

1. A collection of stamps.

2. A collection of toys.

3. A collection of books, etc.

In the same way, we have different types of groups made for different activities.

Such as:

1. A group of boys playing hockey.

2. A group of girls playing badminton.

3. A group of students going for picnic, etc.

In mathematics, a collection of particular things or a group of particular objects is called a set.

26.2 Definition Of A Set

A set is a collection of well-defined objects.

Meaning of "Well-Defined"

Well-defined means, it must be absolutely clear which object belongs to the set and which does not.

For example:

(i) A collection of "Lovely Flowers" is not a set because the objects (flowers) to be included are not well-defined.

Reason: The word "Lovely" is a relative term. What may be lovely to one person may not be so to another person.

(ii) A collection of "Red Flowers" is a set because every red flower will be included in this set, i.e. the objects of the set are well-defined.

(iii) A group of "Young Players" is not a set, as the range of the age for young players is not given. i.e. it cannot be decided which player is to be considered young, i.e. the objects are not well-defined.

(iv) A group of "Players with ages between 14 years and 18 years" is a set, because the range of age of the players is given and so that it can be easily decided which players are to be included and which are to be excluded.

Hence, the objects are well-defined.

Example 1:

State, giving reason, whether or not the following objects form a set:

(i) All the problems in this book that are difficult to solve.

(ii) All the problems in this book that are difficult for Mohit to solve.

(iii) All objects heavier than 28 kg.

Solution:

(i) The given objects do not form a set.

Reason: Some problems may be difficult for one person but not so difficult for others. Hence the given objects are not well-defined.

Thus, the given objects do not form a set.

(ii) The given objects form a set.

Reason: It can be easily found which problems are difficult to solve for Mohit and which are not.

(iii) The given objects form a set.

Reason: Every object can be compared with certainty in relation to weight of 28 kg. Thus, it is very easy to select objects which are heavier than 28 kg, i.e. the objects are well-defined, so they form a set.

The members (objects) of each of the following collections form a set:

(i) students in a class room,

(ii) books in your school bag,

(iii) counting numbers between 10 and 20,

(iv) the students of your class, that are taller than you and so on.

Teacher's Note

Understanding sets helps us organize and classify information in everyday life, from grouping students by grade level to organizing items in a store by category.

Exercise 26(A)

1. State whether or not the following elements form a set; if not, give reason:

(i) All the easy problems in your text book.

(ii) All the three sided figures.

(iii) The first five counting numbers.

(iv) All the tall boys of your class.

(v) The last three days of a week.

(vi) All triangles that are difficult to draw.

(vii) The first three letters of the English alphabet.

(viii) All tasty fruits.

(ix) All the clever boys of class 6.

(x) All the good schools in Delhi.

(xi) All the girls in your class whose heights are less than your height.

(xii) All the boys in your class whose heights are more than your height.

(xiii) All the problems in your Mathematics book that are difficult for Amit.

26.3 Ways Of Representing Sets

A set or a collection of well-defined objects may be represented by pictures drawn inside closed figures, like a circle, a rectangle, a square, etc.

For example:

(i) A set of some fruit can be shown as:

(ii) A set of some sports goods can be shown as:

Instead of showing (drawing) these pictures inside a circle, a rectangle, etc., we can also draw them within a pair of curly braces (brackets).

(i) {banana, apple, grapes, mango}

(ii) {cricket bat, tennis racket, football, hockey}

The two sets shown above by pictures can also be expressed by writing their names as shown below:

(i) {banana, apple, grapes, mango}.

(ii) {cricket bat, tennis racket, football, hockey}.

It is convenient and time-saving to write (express) sets in this way.

Thus, the elements of a set can be expressed through pictures or by writing their names inside a pair of curly braces.

26.4 Elements (Or Members) Of A Set

The objects used to form a set are called its elements or its members.

Generally, the elements of a set are written inside a pair of curly (middle) braces and separated by commas. The name of the set is always written in capital letters.

For example:

A = {p, q, r, s, t}

Here, 'A' is the name of the set, whose elements (members) are p, q, r, s and t.

1. The pair of curly braces {} denotes a set.

2. The Greek letter Epsilon 'e' is used for the words "belongs to", "is an element of", etc.

Therefore p e A will be read as "p belongs to set A" or as "p is an element of set A".

In the same way; q e A, r e A, s e A and t e A.

The symbol 'e' stands for "does not belong to" or "is not an element of".

Therefore a e A will be read as "a does not belong to set A" or as "a is not an element of set A".

In the same way b e A, c e A and so on.

Example 2:

If set A = {3, 6, 9, 12, 15, 18}, state whether the following statements are true or false:

(i) 8 e A

(ii) 12 e A

(iii) 15 e A

(iv) 9, 12 e A

(v) 12, 14, 15 e A

Solution:

(i) False: as element 8 does not belong to the given set A.

(ii) False: as element 12 belongs to the given set A.

(iii) True: as element 15 belongs to the given set A.

(iv) True: as elements 9 and 12 both belong to the given set A.

(v) False: as element 14 does not belong to the given set A.

Teacher's Note

Sets help us understand membership and classification - like understanding which students belong to a particular class or which items belong to a particular category in a store.

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