ICSE Class 8 Maths Chapter 01 Sets

Chapter 1

Sets

In previous classes, you learnt the basic concepts of sets. The following topics were covered:

Concept of a set

Methods of describing a set, by listing its members (roster or tabular form) or by using rule method (set builder form)

Finite sets, infinite sets, empty set, universal set

Cardinal number of a finite set

Equal and equivalent sets

Overlapping and disjoint sets

Subsets

Union and intersection of sets; complement of a set

Venn diagrams

Solving problems by using Venn diagrams

In this chapter and the next chapter, we shall review and strengthen these topics and introduce following new ideas:

Proper subsets

Number of subsets of a finite set

Difference of two sets

Combination of operations like union, intersection, complement

Sets

Any well defined collection of objects is called a set. The objects which belong to the set are called its members or elements.

By a well defined collection of objects, we mean that given a set and an object, it should be possible beyond doubt whether the object belongs to the set or not. For example, the collection of all brave students of your school is not well defined, since a student of your school considered brave by one person might be considered coward by another. Thus, such a collection is not a set.

The sets are usually denoted by capital letters A, B, C etc., and the members of a set are denoted by small letters x, y, z etc.

If x is a member of the set A, we write x \(\in\) A (read as 'x belongs to A') and if x is not a member of the set A, we write x \(\notin\) A (read as 'x does not belong to A'). If x and y both are members of the set A, we write x, y \(\in\) A.

Representation Of A Set

A set can be represented by the following methods:

(i) Description method

(ii) Roster method or tabular form

(iii) Rule method or set builder form.

Description Method

In this method, we make a (well defined) description of the elements of the set and this description of elements is enclosed in curly brackets.

For example:

(i) The set of even whole numbers less than 20 is written as {even whole numbers less than 20}.

Note that 0 \(\in\) {even whole numbers less than 20} while 20 \(\notin\) {even whole numbers less than 20}.

(ii) The set of all months of a year is written as {months of a year}.

Roster Method Or Tabular Form

In this method, we list all the members of the set within braces (curly brackets) and separate these by commas.

For example:

(i) The set A of all odd natural numbers less than 15 in the roster form is written as A = {1, 3, 5, 7, 9, 11, 13}.

(ii) The set M of months of a year having less than 31 days in the roster form is written as M = {February, April, June, September, November}.

(iii) The set L of letters in the word 'JODHPUR' in the tabular form can be written as L = {J, O, D, H, P, U, R}.

Remarks

The order of listing the elements in a set can be changed.

Thus, the set {3, 5, 6, 9} may also be written as {3, 6, 9, 5} or {6, 3, 9, 5} etc.

If one or more elements of a set are repeated, the set remains the same.

Thus, the set {a, b, c, b, b, a} is the same as {a, b, c}.

Each element of a set is listed once and only once, repetitions are removed.

Thus, the set A of letters in the word 'PROFESSOR' is written as A = {P, R, O, F, E, S}

If the number of elements in a set is very large, then we can represent the set by writing a few members which clearly indicate the structure of the elements of the set followed (or preceded) by three dots '...' and then writing the last element (if it exists).

Thus, the set A of even natural numbers between 50 and 500 in the tabular form is written as A = {52, 54, 56, ... , 498}

The set B of odd integers less than 9 in the roster form is written as B = {..., -5, -3, -1, 1, 3, 5, 7}

Rule Method Or Set Builder Form

In this method, we write a variable (say x) representing any member of the set followed by a property satisfied by all element of the set, and enclose it in braces.

If A is the set consisting of elements x having property p, we write A = {x | x has property p}

which is read as "the set of elements x such that x has property p'. The symbol '|' stands for the words 'such that'. Sometimes, we use the symbol ':' in place of the symbol '|'.

For example:

(i) The set A of all odd natural numbers less than 15 in the set builder form is written as A = {x : x = 2n - 1, n \(\in\) N and n \(\leq\) 7}

or A = {x : x = 2n + 1, n \(\in\) W and n \(\leq\) 6}.

(ii) The set P = {0, 3, 6, 9, 12, 15, 18} in the set builder form is written as P = {x : x = 3n, n \(\in\) W and n < 7}.

(iii) The set S = {9, 16, 25, 36, 49, 64, 81, 100} in the set builder form can be written as S = {x | x = n\(^2\), n \(\in\) N and 3 \(\leq\) n \(\leq\) 10}.

Teacher's Note

When organizing a sports team roster, coaches use sets to group players by position or skill level - this is like using the roster method to list all members of a set, where each player belongs (or doesn't belong) to specific team groups.

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