ICSE Class 8 Maths Sets Chapter 01 Sets

Unit One: Sets

Topics Covered

Sets

Operations on Sets

Venn Diagrams

Let's Recap

1. State the type of the following sets, based on their cardinal numbers.

(i) A = {x | x is a consonant in the word AREA}

(ii) B = {x | x = \(\frac{3a}{2}\), a ∈ W}

(iii) C = {x | x = 3x - 12, x ∈ W}

(iv) D = {x | x = 3y + 4, -30 < y < 30}

(v) E = {x | x is a prime number}

(vi) F = {x | x is an odd number, \(\frac{x}{2}\) = 0}

(vii) G = {x | x is a composite number, x < 1000, x ∈ Z}

2. Write all the possible subsets of the following sets:

(i) A = {-5, -4, -3}

(ii) B = {x | x is a letter in the word GNOME}

(iii) C = {x | x is a prime factor of 210}

(iv) D = {x | x is a multiple of 6, x ≤ 24, x ∈ N}

3. If ξ = {x | x is an odd number, 30 < x < 50}, then write the complements of the following subsets:

(i) A = {31, 33, 39, 43, 47}

(ii) B = {x | x = 3a, 10 < a < 16, a is an odd number}

(iii) C = {x | x a prime number, 30 < x < 50}

Dialogue: A teacher and student discuss maintaining a member database for subscription fees, noting that defaulters form a complement set, and discussing how set operations can be performed on computers.

Teacher's Note

Sets help us organize information just like a membership database organizes club members - some pay (main set) and some don't (complement set).

Section 1: Sets

Key Concepts

Roster Method

Set-Builder Method

Cardinal Number of a Set

Finite and Infinite Sets

Singleton and Null Sets

Equivalent and Equal Sets

Disjoint and Joint Sets

Subset, Superset and Universal Set

Proper Subset

Power Set

Complement of a Set

Replacement Set

Introduction

Let us first recall what was learnt about sets and the different types of sets in our previous classes to reinforce our understanding of sets.

Definition of a Set

A set is a well-defined collection of distinct objects. The objects in a set, known as its elements, are enclosed within a pair of braces and the set is denoted by a capital letter of the English alphabet.

A = {1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210}

The elements in set A above are all factors of the number 210.

(i) 15 is an element in set A and this is denoted as 15 ∈ A or 15 belongs to set A.

(ii) 22 is not an element in set A and this is denoted as 22 ∉ A or 22 does not belong to set A.

(iii) If 210 is a multiple of a certain natural number, we can say with certainty that the number is an element of set A. Thus, the elements of set A are well-defined.

(iv) No element is repeated in set A. Thus, the elements of set A are distinct.

Sets of Numbers

We have learnt about different types of numbers in number systems in our previous classes. Although there are infinite numbers of each type, each number of a particular type of numbers is distinct and well-defined.

Set N = The set of natural numbers = {1, 2, 3, 4, 5, 6, ...}

Set W = The set of whole numbers = {0, 1, 2, 3, 4, 5, ...}

Set Z = The set of integers = {..., -3, -2, -1, 0, +1, +2, +3, ...}

Set Q = The set of rational numbers of the type \(\frac{p}{q}\), where p and q ∈ Z, q ≠ 0

= {..., \(-\frac{3}{4}\), ..., -0.2, ..., 0, ..., +0.3125, ..., \(+\frac{1}{2}\), ...}

Set R = The set of real numbers = {..., -3, ..., \(-\frac{6}{7}\), ..., 0, ..., \(\sqrt{2}\), ..., 3.78, ..., ...}

Let us consider the following example.

Name the smallest set of numbers to which \(\sqrt{4}\) belongs to.

\(\sqrt{4}\) = \(\sqrt{2 \times 2}\) = 2 which is a natural number (N). 2 is also a rational number (Q) as it can be expressed as \(\frac{2}{1}\), where 2 and 1 are integers and 1 ≠ 0.

Being a rational number, 2 is also a real number (R). As Q ⊂ R, Z ⊂ Q, and N ⊂ Z, the smallest set that \(\sqrt{4}\) belong to is the set of natural numbers (N).

Set Representation

Roster Method

In the Roster method, also known as the listing or the tabular method, the elements of the set are listed individually, enclosed within braces.

A = {n, v, r, m, t},

B = {8, 16, 24, 32, 40, ...}, and

C = {66, 77, 88, 99}

are examples of set representation by the Roster method.

Set-Builder Method

In the set-builder method, also known as the Rule method, the elements of the set are described by a statement summarising their common property, which is enclosed within braces.

A = {x | x is a consonant in the word ENVIRONMENT},

B = {x | x = 8a, a ∈ N}, and

C = {x | x = 11a, 6 ≤ a ≤ 9, a ∈ N}

are examples that represent the same sets represented by the Roster method earlier, now represented by the set-builder method.

Remember: As the elements of a set need to be distinct, the repetition of elements is not allowed. The set remains unchanged, even if the order of listing the elements is changed. Set A = {0, 1, 2, 3} = {1, 0, 3, 2}

Example 1: Are the following collections sets?

(i) A = {x | 12 < x < 13, x ∈ Q}

(ii) B = {x | x is a student in class VIII of St Mary's School who can swim}

(iii) C = {x | x is a student in class VIII of St Mary's School who is a good swimmer}

(iv) D = {x | 12 < x < 13, x ∈ N}

(i) Although infinite rational numbers lie between 12 and 13 on the number line, each number is distinct and the collection is well-defined. So, A is a set.

(ii) All students of the particular class who can swim will belong to this collection. Thus the collection is well-defined. As each of these students are different from one another, each element is distinct. Thus B is a set.

(iii) Whether a swimmer is 'good' or not is a matter of personal opinion. Thus it cannot be stated with certainty if a particular student will be in this collection or not. Thus this collection C, being not well-defined, is not a set.

(iv) As no natural number lies between 12 and 13, there would be no element in collection D. As this fact defines collection D and as there are no elements in D, there is no chance of repetition of elements. Hence D, although empty, is a set.

Cardinal Number of a Set

The cardinal number of a set is the number of distinct elements in it.

The cardinal number of set A is denoted by n(A).

Example 2: Write the cardinal numbers of sets A and B.

(i) Set A = {0.1, 0.01, 0.001, 0.0001}

There are 4 elements in set A.

Thus, n(A) = 4

(ii) Set B = {x | x = -1^n, n ∈ N}

Now (-1)^(even number) = +1 and (-1)^(odd number) = -1. Thus, x can assume only 2 values, +1 or -1. Hence, n(B) = 2

Example 3: Represent set A = {x | x = 3a - 4, a ∈ N} by the Roster method and find its cardinal number.

When a = 1, x = (3 × 1) - 4 = -1

When a = 2, x = (3 × 2) - 4 = 2

When a = 3, x = (3 × 3) - 4 = 5

Thus, beginning from -1, the value of x increases by 3.

Thus A = {-1, 2, 5, 8, 11, 14, ...}

As there are infinite elements in set A, n(A) = ∞.

Try this! Write the cardinal number of the following set. set A = {1, 3, 5, 7, 9, 11}

Teacher's Note

Cardinal numbers help us count elements in sets, similar to how a teacher counts present students in a classroom each day.

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