Set Theory Chapter Notes | Mathematics Class 6 ICSE PDF Download

Introduction

Imagine organizing your favorite toys, books, or even snacks into groups based on specific rules, like "all red toys" or "all chocolate snacks." This is what set theory is all about! It’s a fun and logical way to group things that share something in common. In mathematics, set theory helps us understand collections of objects clearly and systematically. Whether it’s numbers, letters, or even months of a year, sets make it easier to study and work with groups. Let’s dive into this exciting world of sets and discover how they help us organize and understand collections!

Set

• A set is a collection of well-defined objects, where it’s clear whether something belongs to the set or not.
• Each object in a set is called a member or element.
• Well-defined means there’s no confusion about what’s included in the set.
• If an object appears more than once in a collection, it’s counted only once in a set.
• Sets are denoted by capital letters like A, B, or C, and elements by small letters like x, y, or z.
• The symbol ∈ means "belongs to," and ∉ means "does not belong to."

Representation of Sets

• Sets can be represented in two ways: roster form and set-builder form.

Roster or Tabular Form Method

• In this method, all elements of the set are listed inside curly braces {}, separated by commas.
• It’s a straightforward way to show all members of a set.

Rule or Set-Builder Form Method

• In this method, a rule or property describes all elements of the set, written inside curly braces.
• The format is {x | condition on x} or {x : condition on x}, where | or : means "such that."
• It describes the common property that all elements must satisfy.

Important sets in both forms:

• Natural numbers (N):
  • Roster form: N = {1, 2, 3, 4, …}
  • Set-builder form: N = {x | x is a natural number}
• Whole numbers (W):
  • Roster form: W = {0, 1, 2, 3, …}
  • Set-builder form: W = {x | x is a whole number}
• Integers (Z or I):
  • Roster form: Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
  • Set-builder form: Z = {x | x is an integer}

Types of Sets

Finite Sets

• A set with a limited number of elements that can be counted is called a finite set.

Infinite Sets

• A set with an unlimited number of elements that cannot be counted is called an infinite set.

Empty Sets

• A set with no elements is called an empty set or null set, denoted by {} or φ.

Cardinality

• Cardinality is the number of elements in a finite set, denoted by n(X) for a set X.
• Cardinality is not defined for infinite sets.
• The cardinality of an empty set is zero, i.e., n(φ) = 0.

Solved Examples