Introduction of a Set || What is a Set? || Founder of a set || Sets and their Representation

Introduction of a Set

Modern mathematics is fundamentally based on the idea of the set. Today, almost every area of mathematics makes use of this concept.

• The concept of relations and functions is frequently defined in terms of sets.
• The study of pure mathematics, probability, counting, sequence, geometry, etc.
• It also helps in the solution of several mathematical confusions, both simple and complicated.

Founder of a set (George Cantor)

German mathematician George Cantor $(1845$ $-1918)$ created the theory of sets in $1872. While working on "problems of trigonometric series," he first came across sets.

Sets and their Representation

• In everyday language, we regularly refer to a grouping of related items,

Such as
A group of people, a cricket team, a deck of cards

In Mathematics also come across collections

For example

Whole numbers, line segments, prime numbers, rational numbers, real numbers, etc. are examples of natural numbers.

More specially, we examine the subsequent collections

i)     Odd numbers less than $10$ i.e, $1,$ $3,$ $5,$ $7,$ $9$
ii)    ii) The English alphabet's vowels, such as $a$, $e$, $i$, $o$, and $u$.
iii)    Prime factors of $210$, namely $2$, $3$, $5$, $7$
iv)    The solutions of the equation  $x^2$ $-5x$ $+6$ $=0$, viz $2$ and $3$
v)     various kinds of triangles

Definition of a Set

• A collection of well-defined and distinct objects is called a Set.
• A set is a collection of well-defined objects which are distinct from each other is called a Set.
  
  

Well-defined

It implies that a rule that determines whether or not an object is a member of a set can be expressed.

Some examples which are not well-defined

i)      The collection of brave boys
ii)     The collection of intelligent students.
iii)    The collection of honest humans.
iv)    The collection of favorite places.
v)     The collection of tasty dishes.

In the above examples, we can examine that the words brave, intelligent, honest, favorite, and tasty are not well-defined because.

• A boy may be brave for one person but may not be for another.
• A student may be intelligent for one person but may not be for another.
• A human may be honest for one person but may not be for another.
• A place may be a favorite for one person but may not be for another.
• A dish may be tasty for one person but may not be for another.

Only the same kind of objects will be presented in the set

• A bottle cannot be included in the set of books.
• A cat cannot be included in the set of mobiles.

Examples of well-defined

i)    The collection of boys.

ii)   The collection of numbers.

iii)  The collection of places.

iv)  The collection of dishes.

v)   The collection of blue pens.

Distinct

• To be distinct, an object must be distinct from all other objects in the set.
• A member of a set cannot be repeated.

Examples

The set of letters of the word "Pakistan".

may be written as:

$\{p,$ $a,$ $k,$ $i,$ $s,$ $t,$ $n\}$

The alphabetical grouping of the word "Mathematics."

$\{m,$ $a,$ $t,$ $h,$ $e,$ $i,$ $c,$ $s\}$