ICSE Class 6 Maths Chapter 27 Types of Sets

Chapter 27: Types of Sets

Introduction

In this chapter, we shall be finding whether:

(i) a given set has a countable number of elements or not,

(ii) a given set is empty or not,

(iii) two given sets have identical elements or not,

(iv) two given sets have an equal number of elements or not,

(v) two given sets have some elements in common or not, etc.

Finite Set

A set is said to be a finite set if it has a limited number of elements, i.e. the number of elements in it can be counted.

For example:

1. P = {Natural numbers less than 50} = {1, 2, 3, ....., 49}. Set P is finite as it has a countable number of elements.

2. Q = {x : x is a student in your school}

3. R = {Whole numbers between 5 and 45}

and so on.

Infinite Set

A set is said to be infinite if it has an unlimited number of elements i.e. the number of elements of such a set cannot be counted.

For example:

1. Q = Set of whole numbers = {0, 1, 2, 3, .... }. Each of these sets has an uncountable number of elements.

2. R = {Stars in the sky}

3. A = {x : x is a natural number greater than 32} = {33, 34, 35, 36, ..... }

and so on.

Such sets are expressed in the Roster Form by writing a few elements and then putting some dots to show that the elements continue till infinity.

The Empty Set or The Null Set

It is a set that has no elements.

The empty set is expressed by a pair of curly braces with no element written inside them, i.e. { } represents an empty set.

The empty set (the null set) is also represented by the Greek letter Ø, spelt as phi. The symbols { } and Ø represent the same set, i.e. { } = Ø.

The empty set is a finite set.

Examples:

(i) Let A = {Triangles with four sides} then A = Ø, i.e. A = { }. There is no triangle with four sides.

(ii) Let B = {Natural numbers less than 1} then B = { }, i.e. B = Ø. There is no natural number less than 1.

Equal Sets

Two sets are said to be equal if the elements of the two sets are the same, i.e. the elements of the two sets are identical. The symbol used for equality of sets is the usual sign "=", i.e. "equal to".

For example:

Let A = {1, 2, 3, 4} and B = {Natural numbers less than 5}. Then, set A is equal to set B.

And so we write: Set A = Set B, or simply A = B

Equivalent Sets

Two sets are said to be equivalent if the number of elements in both the sets are equal. The elements may be the the same or they may be different, but each set must contain the same number of elements.

For example:

Consider A = {x, y, z} and B = {Patna, Calcutta, Delhi}. Here, A and B are equivalent sets because they have an equal number of elements (both A and B have three elements).

1. Equal sets are equivalent but the converse is not always true.

2. Two empty sets are always equal.

3. Two infinite sets are always equivalent.

Disjoint Sets

If two given sets have no element in common, they are disjoint sets.

For example:

(i) Let A = Set of students of Class X and B = Set of students of Class XII. Since, no student can be common to the two classes, sets A and B are disjoint.

(ii) Let P = {a, b, c, d} and Q = {1, 2, 3, 4, 5}. Clearly, sets P and Q have no element in common; therefore, sets P and Q are disjoint sets.

Overlapping Sets

If two given sets have at least one element in common, they are said to be overlapping sets.

Overlapping sets are also known as joint sets.

For example:

(i) If set A = {5, 6, 7, 8, 9, 10} and set B = {4, 6, 8, 10, 12}, sets A and B are overlapping as they have elements 6, 8 and 10 in common.

(ii) Let P = Set of students of Class X and Q = Set of students of Class X in Sophia School. Clearly, the students of Class X in Sophia School are common to the two given sets; therefore sets P and Q are overlapping sets.

Teacher's Note

Understanding different types of sets helps us organize information in daily life, such as grouping students by class or separating objects by their properties. These concepts are fundamental to mathematical reasoning and problem-solving.

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