SETS 

Set: A collection of well defined objects is called a set. 

- Objects of collection is called "element" or "member".

Properties: 

(i) All elements of a set should have special property. 

(ii) They all should be differ to each other or they do not repeat. 

(iii) They should be well defined. 

• Name of any set is writen in capital letter of English Alphabets

        A, B, C, D ........................ X, Y, Z

• Elements and members of a set is writen in small letter of English Alphabets or using number system within a brace (medium bracket) with using commas { } 

        Example : A = {a, b, c, d} Or B = {1, 2, 3, 4}

Standard symbols of some special sets:

N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z⁺ : the set of positive integers
Q⁺ : the set of positive rational numbers, and
R⁺ : the set of positive real numbers.

Symbols and their meaning: 

(i) " ∈ "  (epsilon) : "belong to" 

If a is an element of a set A, we say that “ a belongs to A”

in symbolic form we write it as : a ∈ A 

(ii) " ∉ " : "not belong to" 

When any element which not belong to any given set then we use symbol ∉ : "not belong to" 

 if a is not an element of set A. we say that "a not belong to A"

in symbolic form we write it as : a ∉ A 

(iii) " ⇒ " : emplies 

In common language "⇒" means "emply toward ..... also ......" 

(iv) " = " : Equal to

A = B, it means set A is equal to set B. 

(v) " ≠ " : Not equal to 

A ≠ B, It means set A not equal to set B. 

(vi) “⇔” is a symbol for two way implications, and is usually read as " if and only if ". 

(vii) " : " : colon Here in the set colon stands for "such that" 

Representation of sets: 

Every set is introduced by its elements. So for expressing a set before we have to express its elements. 

Now, Method to express the elements of a set. 

(1) Roster Or tabular form : 

In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example,

(i) First five natural numbers: 

A = {1, 2, 3, 4, 5}. 

(ii) First four letter of English Alphabets

B = {a, b, c, d}.

(iii) The set of all natural numbers which divide 42

C = {1, 2, 3, 6, 7, 14, 21, 42}.

(iv) The set of all vowels in the English alphabet.

D = {a, e, i, o, u}.

Note: 

In Roster Form no-element is generally repeated

Example: 

The set of letters forming the word 'ELEMENT'

E = {E, L, M, N, T} 

Here no-element has been repeated. 

(2) Set-builder form 

In set-bulder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set.

Example: 

D = { a, e, i, o, u} 

All elements of set A have a common property e.i vowel in English alphabet. 

Then, in set-builder form:

D = { x : x is a vowel in english alphabet } 

This set will be read as :

" the set of all x such that x is a vowel in english alphabet. "

C = {y : y is a natural number which divides 42}

B = (z : z is a first four letter of english alphabet} 

Other examples 

(i) A = {1, 2, 9, 25}  Roster form

Set-builder form

A = {x : x = n², where n ∈ N and n < 6} 

(ii) B = {P, R, I, N, C, A, L}

Set-builder form

B = {x : x is a letter of the word PRINCIPAL}

(iii) C = {1, 2, 3, 6, 9, 18}

Set-builder form 

C = { x : x is a positive integer and is a divisor of 18}

(iv) D = (3, 6, 9, 12}

Set-builder form 

D = {x : x = 3n where n ∈ N and 0 < n < 5}