Set Theory

 

  • Introduction

The set theory, developed by George Cantor, forms a basis for advanced mathematical topics, such as the calculus, real and complex analysis etc. It defines functions, which are an extremely important entity in mathematics.

  • Definition

A set is a collection of well defined objects. For example, the set of all subjects studied by you in class 10. This includes English, Marathi, Hindi, French, Mathematics, Science and Social Science.

The collection of all intelligent people in your locality does not qualify as a set. The intelligence cannot be quantied and therefore, it is not a well-defined object. The set of all prime numbers less than 15 is another example. This set includes 2,3,5,7,11 and 13.

The objects which belong to a set are called elements of that set. The repetition of elements is NOT allowed in set theory. So, if there are 2 elements, which are identical, they will be treated as a single element.

  • Representation of Set

A set is represented by capital Roman alphabets, such as A, P etc. It is customary to write down the elements of set within curly brackets. So, the set of all prime numbers less than 15 will be written as (let us call the set as P)

{P = \{ 2, 3, 5, 7, 11, 13\} }

This representation of a set, where we list down all the elements, is known as the Roster form. The same set can also be written in Set-builder form as follows:

{P = \{ x \ | \ x \ is \ a \ prime \ number \ less \ than \ 15 \} }

  • Sets of Numbers

The set of natural numbers, also known as counting numbers, is denoted by

{\mathbb {N} = \{ 1, 2, 3, \cdots \} }

The set of whole numbers is denoted by

{\mathbb {W} = \{0, 1, 2, 3, \cdots \} }

Thus, {\mathbb {W}} is made up of the number zero as well as { \mathbb {N}}.

The set of integers is denoted by

{\mathbb {Z} = \{\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots \}}

The next set of numbers is the set of rational numbers. These numbers can be represented in the form {\frac p q} , where {p} and {q} are any integers and {q \ne 0}. It is denoted by {\mathbb {Q}}.

There are numbers, such as {\sqrt 2, \sqrt 3, e}, which cannot be represented in the form {\frac p q} . Such numbers are known as irrational numbers. The set of irrational numbers is denoted by {\mathbb I}.

When the set of rational numbers {\mathbb {Q}} and the set of irrational numbers {\mathbb I} are combined together, the set of real numbers, {\mathbb R} is obtained.

  • Terminology

When all the elements of set {A} and set {B} are identical, we say that {A} and set {B} are equal. So,

{A = B}

In roster form, equal sets are identical; but their set-builder forms may differ. Note that they are one and the same.

An empty set (or a null set) is the one, which contains no element. It is denoted by

{\{ \ \} \ or \ \phi}

For example, {\{x | x \in \mathbb N \ and \ 4<x<5  \}}.

A set with only 1 element in it is known as singleton set. For example, {\{x | x \in \mathbb N \ and \ 4<x<6  \}}. This contains only 1 element, which is {5}.

Set {A} is said to be a subset of set {B}, when all elements in set {A} are contained in set {B}. Set {B} is then called the superset of {A}. We denote this by,

{A \subseteq B \ and \ B \supseteq A}

For example, {A = \{1, 2, 5, 6 \}, \ B = \{-3,0,1,1.5,2,4,5,6,8\} }. Notice that {B} contains all elements which are there in {A}.

A non-empty set {P} is said to be a proper subset of a set {Q}, such that there exists at least one element in {Q}, which does not belong to {P}. i.e.

{P \subset Q, \ when \ \exists \ x,\  x \in Q, \ x \ne P}

A set, which has a finite number of elements is called a finite set. e.g.

{P = \{ x | x \in \mathbb N, \ x < 6}

A set, where the process of counting elements does not end is known as an infinite set. e.g. the set of natural numbers, {\mathbb N}.

A universal set is the one, which in a discussion, serves as a parent set. In other words, all other sets are subsets of this set. Note that, for all kinds of numbers we discussed earlier, the set of real numbers, {\mathbb R} is the universal set. Universal set is generally denoted by {U}.

  • Operations on sets

  • Union

Union of 2 sets {A} and {B} is the set of all elements, which are present either in {A} or in {B} or in both {A} and {B}. It is denoted by

{A  \cup B}

  • Intersection

Intersection of 2 sets {A} and {B} is the set of all elements, which are common to both A and B. It is denoted by

{A \cap B}

  • Complement of a set

Let {A} be any set. The complement of set {A} is the set of all elements, which do not belong to {A}; but are contained in the universal set {U}. It is denoted by {A^c} or {A^0}.So,

{A \cap A^c = \phi , \ A \cup A^c = U}

  • Power set

The power set of a set {A} is the set of ALL subsets of {A}, including the null set and {A} itself. So, if {A = \{ 2,4,5 \}}, the power set will be

{P = \{ \ \{ \ \}, \{2 \}, \{4 \}, \{5 \}, \{2,4 \}, \{2,5 \}, \{ 4,5 \}, \{ 2,4,5 \} \ \}}

Note that if a set has {n} elements, its power set will have {2^n} elements. So, power set of A will have {2^3=8} elements.

  • Venn Diagrams

A Venn diagram of a set is the pictorial representation of the set. Generally, Venn diagrams are geometrical figures, such as circle, triangle, ellipse etc. The operations on set can be represented readily using these diagrams.

  • De Morgan’s laws

{(A \cup B)' = A' \cap B'}

{(A \cap B)' = A' \cup B'}

  • Few More Terms

  • Commutativity of Operations:

{A \cup B = B \cup A }

{A \cap B = B \cap A}

  • Associativity of Operations:

{A \cup (B \cup C )= (A \cup B) \cup C}

{A \cap (B \cap C )= (A \cap B) \cap C}

  • Idempotent laws

{A \cap A = A, A \cup A = A}

  • Distributive Properties of Operations

{A \cap (B \cup C) = (A \cap B) \cup (A \cap C)}

{A \cup (B \cap C) = (A \cup B) \cap (A \cup C)}

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