Set theory | Introduction

Eleven Standard >> Set theory | Introduction

Introduction to Set Theory

A set is a well-defined collection of distinct objects, considered as a single entity. These items may include numbers, symbols, elements, or even other sets. In set theory, the objects within a set are referred to as its "elements" or "members."

For example:

The set of prime numbers: {2, 3, 5, 7, 11, 13, ...}
The set of colors: {red, blue, green, yellow, ...}
The set of fruits: {apple, banana, grape, ...}

Georg Cantor's development of set theory marked a pivotal advancement in mathematics, transforming how infinity is understood and offering profound insights into the nature and characteristics of sets. Cantor's work laid the foundation for modern set theory and has had a profound impact on various branches of mathematics.

Georg Cantor (1845–1918), a German mathematician, is best known for his pioneering work on the theory of infinite sets. He introduced revolutionary ideas that challenged traditional notions of size and counting. Some of Cantor's key contributions include:

Cardinality of Sets: Cantor introduced the concept of "cardinality" to compare the sizes of sets. Two sets possess equal cardinality when there exists a one-to-one correspondence (bijection) connecting their elements. He showed that some infinite sets, such as the natural numbers (1, 2, 3, ...), have the same cardinality as proper subsets of themselves, defying the intuition that infinite sets must always be larger.

Cantor introduced a classification of sets based on their cardinality, dividing them into countable and uncountable types. A set is considered countable if its elements can be matched one-to-one with the natural numbers. In contrast, sets that cannot be paired this way are called uncountable. For instance, the set of natural numbers is countable, whereas the real numbers between 0 and 1 form an uncountable set.

Diagonal Argument: Cantor famously used his diagonal argument to prove that the real numbers are uncountable. This elegant proof showed that no matter how one attempts to list all the real numbers between 0 and 1, there will always be real numbers missing from the list.

Transfinite Numbers: Cantor introduced the concept of "transfinite numbers" to extend the notion of infinity. He defined different levels of infinity, such as aleph-null (ℵ₀) for countable infinity and larger aleph numbers for uncountable infinity. This concept opened up new avenues for understanding the hierarchy of infinities.

Set Operations: Cantor also developed foundational concepts and techniques for performing operations on sets, such as union, intersection, and power sets. He introduced a structured approach to handling and examining sets through set operations.

Set notations:

Set notation uses symbols to describe sets and their characteristics within mathematics. It allows us to describe and communicate various aspects of sets, including their elements, relationships, and operations. Here are some commonly used symbols and notations in set theory:

Curly Braces { }: Sets are often enclosed in curly braces. For example:

A set of natural numbers: {1, 2, 3, 4, ...}
A set of even integers: {..., -4, -2, 0, 2, 4, ...}
Element Symbol (∈): The symbol "∈" is used to indicate that an element belongs to a set. For example:

3 ∈ {1, 2, 3, 4, ...} (3 is an element of the set of natural numbers)
π ∉ {1, 2, 3, 4, ...} (π is not an element of the set of natural numbers)
Subset Symbol (⊆): The symbol "⊆" represents subset relation. A ⊆ B indicates that set A is a subset of set B. For example:

{2, 3} ⊆ {1, 2, 3, 4, ...} (The set {2, 3} is a subset of the set of natural numbers)
Proper Subset Symbol (⊂): The symbol "⊂" represents proper subset relation, indicating that a set is a subset of another set but not equal to it. When we say A ⊂ B, it signifies that set A is included within set B, yet A and B are not identical.

Union (∪) and Intersection (∩): These symbols represent set operations. The union of sets A and B, denoted by A ∪ B, includes all elements that belong to either set. The intersection, written as A ∩ B, consists of elements shared by both sets.

Complement ('): The representation of the complement of set A is indicated by A'. It includes all elements from the universal set that are not part of set A.

Empty Set (∅ or {}): The empty set has no elements and is often represented as ∅ or {}.

Cardinality (|A|): The cardinality of a set A, written as |A|, indicates the total number of elements contained in the set. Power Set (P(A) or \(2^A\)): The power set of A, expressed as P(A) or \(2^A\)), refers to the collection of all possible subsets of A. This includes both the empty set and the set A itself. Set-Builder Notation: This notation defines a set by specifying a condition that its elements must meet. For instance, {x | x is an even integer} describes the set of all even integers.
{x ∈ R | \(x^2\) < 10} represents the set of real numbers whose square is less than 10.

Representation of set:

Representing a set involves expressing its elements and characteristics in a structured and understandable way. Here are several ways sets can be represented:

Roster or List Notation:-

This approach involves listing the elements of a set inside curly brackets.
Example: A = {1, 2, 3, 4} represents the set containing the numbers 1, 2, 3, and 4.

Set-Builder Notation:-

This notation describes the set by specifying a rule or condition that its elements must satisfy.
Example: B = {x | x is a prime number less than 10} represents the set of prime numbers {2, 3, 5, 7}.