Set Notation/ Conventions in Set || Sets in Mathematics World || Order of a finite set/ Cardinal number of a set || Methods to write a Set

Set Notation/ Conventions in Set

•  English capital letters are typically used to represent sets.

i.e.,  $A,$ $B,$ $C,...$ $X,$ $Y,$ $Z$

• Small letters from the English alphabet are used to represent the elements of a set.

i.e,  $a,$ $b,$ $c,...$ $x,$ $y,$ $z$

• Its elements are written within curly brackets $\{$  $\}$ and each element is separated by commas $($ $,$ $)$.
• The phrases objects, elements, and members of a set are similar.
• If $a$ is an element of a set, we say that "$a$ belongs to set $A$" or "$a$ is a member of set $A$". The Greek symbol $ε$ (epsilon) is used to indicate the phrase "belongs to" Thus, we write $a\epsilon A$
• If "$b$" is not an element of the set "$A$", we write $b∉A$ and read as "$b$ is not an element of the set $A$".
  
  

Sets in Mathematics World

$N$:    The set of all natural numbers.
$W$:   The set of all whole numbers.
$Z$:    The set of all integers.
$Q$:    The set of all rational numbers.
OR
$Q$ $=\{x|x$ $=\frac{p}{q}$ $:p,q$ $\varepsilon Z$ $,q\neq 0\}$

$Q'$: The set of all irrational numbers.
OR
$Q'$ $=\{x|x$ $\neq \frac{p}{q}$ $:p,q\varepsilon Z,$ $q\neq 0\}$

$R$:  The set of all real numbers.
$R$ $=Q∪Q'$

Cardinal number of a set or Order of a finite set.

• The number of all elements in a finite set $"A"$ is called the order of this set.
•  It is denoted by $O(A)$ or $n(A)$.
•  It is also called a cardinal number of the set.
•  It is also called a cardinal number of the set.

Example:
$A$ $=\{a,$ $b,$ $c,$ $d\}$                                   $⇒$    $n(A)$ $=4$
$B$ $=\{2,$ $4,$ $6,$ $8,$ $10,$ $12,$ $14\}$              $⇒$    $n(B)$ $=7$
$C$ $=\{1,$ $2,$ $3,...$ $98,$ $99,$ $100\}$             $⇒$    $n(C)$ $=100$

Methods to write a Set

There are three methods to write a set

1)    Descriptive Form
2)    Tabular Form/ Roster Form
3)    Set Builder Form

Descriptive Form

• Descriptive form refers to a collection that may be described with the aid of a statement.

Example:
$A=$ The collection of all real numbers under $10.
$B=$ The English alphabet's first five letters set.
$C=$ The set of positive integers.

Tabular Form/ Roster Form

• In roster form, all the elements of a set are listed and denoted by curly braces $\{$ $\}$. and commas $($ $,$ $)$ to separate the two.

Example:

In roster form, the collection of all dice numbers is represented as

$C$ $=\{1,$ $2,$ $3,$ $4,$ $5,$ $6\}$

Points to be noted in the roster form

• The arrangement of the elements on a roster is not necessary.

Example:
You can write all of the vowels in the English alphabet as
$\{a,$ $e,$ $i,$ $o,$ $u\}$    or    $\{a,$ $i,$ $e,$ $o,$ $u\}$    or    $\{e,$ $i,$ $a,$ $u,$ $o\}$    or    $\{u,$ $a,$ $e,$ $i,$ $o\}$

• A set is indefinitely long if there are three dots at the end of it.
  
  

Example:

$\{1,$ $3,$ $5,...\}$ is used to represent the set of all odd natural numbers.

• An element is not repeated in the roster format.

i.e., Each element is considered to be distinct.

Example:
The set of letters that make up the word "school" is
$\{s,$ $c,$ $h,$ $o,$ $l\}$

Set Builder Form

• This form asks us to list a property or rule that provides us with all of the set's components.
• $A$ $=\{x: P(x)\}$ where $P(x)$ is the property by which $xεA$ and colon $":"$ stands for "such that".

Example: 
$A$ $=\{1,$ $2,$ $3,$ $4,$ $5\}$
$\{x:x\le 5,$ $x\epsilon N\}$    $→$  Set Builder

$B$ $=\{0,$ $1,$ $8,$ $27,$ $64\}$
$B=\{x^3$ $:$ $x<5,$ $xεW\}$

$C$ $=\{2,$ $4,$ $6,$ $8,$ $10\}$
$C$ $=\{x:x\le 10$ $\wedge  x\epsilon E\}$