ICSE Class 8 Maths Sets Chapter 01 Idea of Sets

Idea of Sets

Sets

You are already familiar with some basic concepts of sets. Let us review what you have learnt and then move on to new concepts.

A set is a collection of well-defined, distinct objects. The objects of a set are called members or elements of the set. We call a set a "well-defined collection of objects" because we can decide (with absolute certainty) whether a given object is a member of the set.

Examples

(i) The set of all integers

(ii) The set of vowels of the English alphabet

(iii) The set of rivers of India

(iv) The set of students of class VIII of your school

We usually denote a set by a capital letter of the English alphabet, such as A, B, C, X, Y and Z and its elements, by small letters of the English alphabet, such as a, b, c, x and y.

If a is an element of the set X, we write \(a \in X\) and read this as "a belongs to the set X". If x is not an element of the set A, we write \(x \notin A\) and read this as "x does not belong to A".

Representation of Sets

A set is usually represented in the tabular form or the set-builder form.

Roster Method or Tabular Form

In this method, we represent a set by listing all its elements between braces.

Examples

(i) The set A of the vowels of the English alphabet is represented as \(A = \{a, e, i, o, u\}\).

(ii) The set X of the days of a week is represented as \(X = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}\).

The order in which we list the elements of a set within braces is immaterial. Thus, each of the following denotes the same set.

\(\{a, b, c\}, \{b, a, c\}, \{a, c, b\}, \{c, a, b\}, \{c, b, a\}, \{b, c, a\}\)

In listing the elements of a set in the tabular form, we do not repeat any element. Thus, if B is the set of digits in the number 15,312,142 then \(B = \{1, 5, 3, 2, 4\}\).

Rule Method or Set-Builder Form

We can represent a set by stating a property which its elements satisfy. Thus, the set of natural numbers less than 10, is

\(A = \{x | x \text{ is a natural number, } x < 10\}\)

or \(A = \{x : x \text{ is a natural number and } x < 10\}\).

We read this as "A is the set of all elements x such that x is a natural number and x is less than 10."

Examples

(i) The set \(X = \{2, 4, 6, 8\}\) can be written in the set-builder form as \(X = \{x | x \text{ is an even natural number and } x \leq 8\}\).

(ii) The set of prime numbers that are less than 15 can be written in the set-builder form as \(A = \{x | x \text{ is a prime number and } x < 15\}\).

The set A can be written in the tabular form as \(A = \{2, 3, 5, 7, 11, 13\}\).

We sometimes represent a set by describing a property of its elements inside braces.

Examples

\(A = \{\text{days of a week}\}; B = \{\text{one-digit odd numbers}\}\)

In the tabular form, \(A = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}\)

\(B = \{1, 3, 5, 7, 9\}\)

Some Special Sets

(i) \(N = \{1, 2, 3, 4, \ldots\}\), (set of natural numbers)

(ii) \(W = \{0, 1, 2, 3, \ldots\}\), (set of whole numbers)

(iii) \(Z = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\), (set of integers). This is also denoted as I.

Solved Examples

Example 1

Write each of the following sets in the roster form.

(i) The set of the months in a year that end with y

(ii) The set of the months in a year that have 31 days

(iii) The set of odd numbers between 10 and 20

Solution

(i) \(\{\text{January, February, May, July}\}\)

(ii) \(\{\text{January, March, May, July, August, October, December}\}\)

(iii) \(\{11, 13, 15, 17, 19\}\)

Example 2

Express each of the following sets in the set-builder form.

(i) The set of prime numbers between 20 and 30

(ii) The set of whole numbers which are divisible by 5 and are less than 35

(iii) The set of the factors of 25

Solution

(i) \(\{x : x \text{ is a prime number, } 20 < x < 30\}\)

(ii) \(\{x : x = 5n, n \in W \text{ and } n < 7\}\)

(iii) \(\{x | x \text{ is a factor of } 25\}\)

Example 3

Write the following sets in the roster form.

(i) \(\left\{x : x = \frac{n}{2n+1}, n \in N \text{ and } n < 4\right\}\)

(ii) \(\{x | 5x + 3 < 24, x \in W\}\)

(iii) \(\{x : x = 2r + 3, r \in I \text{ and } -2 < r \leq 3\}\)

Solution

(i) Given, \(n \in N\) and \(n < 4\). So, \(n = 1, 2, 3\). Also, \(x = \frac{n}{2n+1}\).

Substituting \(n = 1, 2, 3\), we get \(x = \frac{1}{3}, \frac{2}{5}, \frac{3}{7}\) respectively.

\(\therefore\) the given set is \(\left\{\frac{1}{3}, \frac{2}{5}, \frac{3}{7}\right\}\).

(ii) Here, \(5x + 3 < 24\). So, \(5x < 21\) or \(x < \frac{21}{5}\).

Since \(x \in W\), we get \(x = 0, 1, 2, 3, 4\). So, the given set is \(\{0, 1, 2, 3, 4\}\).

(iii) Given, \(-2 < r \leq 3\) and \(r \in I\). So, \(r = -1, 0, 1, 2, 3\). Also, \(x = 2r + 3\).

Substituting \(r = -1, 0, 1, 2, 3\), we get \(x = 1, 3, 5, 7, 9\) respectively.

Hence, the given set is \(\{1, 3, 5, 7, 9\}\).

Example 4

Express the following sets in the tabular and set-builder forms.

(i) The set E of even natural numbers

(ii) The set X of the factors of 36

Solution

(i) \(E = \{2, 4, 6, 8, \ldots\}\) [tabular form]

\(= \{x : x = 2n, n \in N\}\) [set-builder form]

(ii) \(X = \{1, 2, 3, 4, 6, 9, 12, 18, 36\}\) [tabular form]

\(= \{x : x \text{ is a factor of } 36\}\) [set-builder form]

Example 5

Express the following sets in the set-builder form.

(i) \(\left\{\frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10}\right\}\)

(ii) \(\{0, 2, 4, 6, 8, 10, 12\}\)

(iii) \(\left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}\right\}\)

(iv) \(\{-40, -35, -30, \ldots, 20\}\)

Solution

(i) \(\left\{x : x = \frac{n}{n+1}, n \in N \text{ and } 4 \leq n \leq 9\right\}\)

(ii) \(\{x | x = 2n, n \in W \text{ and } n \leq 6\}\)

(iii) \(\left\{x : x = \frac{1}{2^m}, m \in N \text{ and } m \leq 6\right\}\)

(iv) \(\{x | x = 5p, p \in I \text{ and } -8 \leq p \leq 4\}\)

Teacher's Note

Sets are used in everyday life when we organize items like groceries by type or group students by grade. Understanding how to describe collections helps us think clearly about what belongs together.

Remember These

1. A collection of distinct objects is a set if one can decide with absolute certainty whether a particular object is a member of the collection.

2. In the roster method (tabular form), the elements of a set are listed between braces {}.

3. In the rule method, the elements are described by a common property of the members.

4. No element of a set is repeated while writing a set by the roster method.

Teacher's Note

The distinction between roster and rule methods mirrors how we describe groups in real life - either by listing members or by stating criteria for membership.

Exercise 1A

1. Which of the following collections are sets?

(i) The collection of the positive integers that are less than 6

(ii) The collection of big cities of India

(iii) The collection of rich people in India

(iv) The collection of the integers that are divisible by 3

2. If A = {1, 2, 3, 4, 5, 6} then which of the following statements are true?

(i) \(6 \in A\)

(ii) \(7 \notin A\)

(iii) \(2, 3 \text{ and } 5 \in A\)

(iv) \(\{1, 2, 3\} \in A\)

(v) \(6 \text{ and } 8 \in A\)

3. Write each of the following sets by the roster method.

(i) The set of the last three months of a year

(ii) The set of the angles of \(\triangle ABC\)

(iii) The set of even integers between 25 and 35

4. Express each of the following sets in the set-builder form.

(i) The set of even integers between 11 and 21

(ii) The set of whole numbers that are divisible by 6 and are less than 48

(iii) The set of the factors of 30

5. Express the following sets in the tabular and set-builder forms.

(i) The set of positive integers

(ii) The set of odd natural numbers

(iii) The set of the factors of 24

(iv) The set of the prime factors of 48

(v) The set of natural numbers which are perfect squares and are less than 50

6. Write the following sets in the tabular form.

(i) \(\left\{x | x = \frac{n}{n+1}, n \in N \text{ and } n \leq 9\right\}\)

(ii) \(\{p : p = 3n+1, n \in W \text{ and } 2 \leq n \leq 5\}\)

(iii) \(\{r : 7r + 5 < 36, r \in W\}\)

(iv) \(\{x : x = u + 3, 10 \leq u \leq 14 \text{ and } u \in N\}\)

(v) \(\{p : p = 7m, m \in I \text{ and } -1 \leq m \leq 3\}\)

(vi) \(\{x | x \in N \text{ and } x^3 < 30\}\)

7. Express the following sets in the set-builder form.

(i) \(\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\right\}\)

(ii) \(\{3, 6, 9, 12, 15\}\)

(iii) \(\{0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40\}\)

(iv) \(\{1, 2, 13, 26\}\)

(v) \(\{-24, -12, 0, 12, 24, 36, 48, 60\}\)

Answers

1. (i) and (iv) are sets

2. (i), (ii) and (iii)

3. (i) \(\{\text{October, November, December}\}\) (ii) \(\{\angle BAC, \angle ABC, \angle ACB\}\) (iii) \(\{26, 28, 30, 32, 34\}\)

4. (i) \(\{x | x = 2n, n \in I \text{ and } 6 \leq n \leq 10\}\) (ii) \(\{x : x = 6p, p \in W \text{ and } p < 8\}\) (iii) \(\{x : x \text{ is a factor of } 30\}\)

5. (i) \(\{1, 2, 3, \ldots\}; \{x | x \in N \text{ and } x > 0\}\) (ii) \(\{1, 3, 5, \ldots\}; \{x : x = 2n - 1 \text{ and } n \in N\}\)

(iii) \(\{1, 2, 3, 4, 6, 8, 12, 24\}; \{x | x \text{ is a factor of } 24\}\) (iv) \(\{2, 3\}; \{x | x \text{ is a prime factor of } 48\}\)

(v) \(\{1, 4, 9, 16, 25, 36, 49\}; \{x : x = n^2, x \in N \text{ and } n \leq 7\}\)

6. (i) \(\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10}\right\}\) (ii) \(\{7, 10, 13, 16\}\) (iii) \(\{0, 1, 2, 3, 4\}\) (iv) \(\{13, 14, 15, 16, 17\}\)

(v) \(\{-7, 0, 7, 14, 21\}\) (vi) \(\{1, 2, 3\}\)

7. (i) \(\left\{x : x = \frac{1}{r}, r \in N\right\}\) (ii) \(\{x | x = 3n, n \in N \text{ and } n \leq 5\}\) (iii) \(\{x : x = 4p, p \in W \text{ and } p < 11\}\)

(iv) \(\{x : x \in N \text{ and } x \text{ is a factor of } 26\}\) (v) \(\{x | x = 12n, n \in I \text{ and } -2 \leq n \leq 5\}\)

Teacher's Note

When students practice converting between roster and rule forms, they develop flexibility in thinking about the same mathematical object in different ways, much like describing a person by their traits versus listing them in a directory.