"A Set is a well-defined collection of distinct objects"

Well-defined means, there must be a rule or rules with the help of which a particular object/element is identified as a member or not a member of the set.

Consider, all days of a week. {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}. This forms a set because any day can be identified as a member of the set.

Consider, 2 days of a week. This is not well defined because the elements of this collection vary from person to person. One person may say {Sunday, Monday}, other person may say {Tuesday, Saturday}, another person may say {Wednesday, Sunday} and so on. So this is not a set.

Consider, 2 days of a week starting with letter T. This is well defined because all days starting with letter T can be definitely identified {Tuesday, Thursday]. So this is a set.

Lets see one more example,
Consider, 4 positive integers. This is not well defined because the elements of this collection vary from person to person. One person may say {1,3,5,6}, other person may say {10, 20, 30, 40}, another person may say {100, 56, 73, 225} and so on. So this is not a set.

Consider, 4 positive integers between 10 and 15. This is well defined because the elements can be definitely identified {11,12,13,14}. So this is a set.

Representing a Set
The sets are generally denoted by capital letter A,B,C....X,Y,Z.
The elements of set are generally denoted by small letters a,b,c...x,y,z.
Described within braces { }
There are two forms of representing sets, (i) Roster form, (ii) Set-Builder form

(i) Roster/ Tabular Form
  • All elements of the set are listed.
  • Elements are separated by commas.
  • Elements are not repeated.
  • The order of the elements listed is immaterial.
(ii) Set-Builder Form
  • Element of set is described using small letter followed by colon :
  • After colon, characteristic property of the elements is written.
Example,
All vowels of English alphabet
Roster form󠀬󠀤󠀤
A={a,e,i,o,u}
Set-Builder form
A={x: x is a vowel in English alphabet}
 
Set of all prime numbers which are divisor of 6
Roster form󠀬󠀤󠀤
A={2,3}
Set-Builder form
A={x: x is a prime number and divisor of 6}

Consider set A={a,e,i,o,u},
it can also be written as, A={a,i,o,u,e} or A={i,o,u,a,e} or A={u,o,i,e,a} so on.
All these are correct because all the elements of set are present, only the order is different and in Roster form order of the elements is immaterial.

 Consider a set of all the letters of the word MATHEMATICS
In this word M,A,T are repeated and in Roster form, elements are not repeated. So the Roster form is,
A={M,A,T,H,E,I,C,S}

∈ (belongs to) and ∉ (doesn't belongs to)
If an element x is in set A, then we say x belongs to set A and is denoted as x∈A.
If an element x is not in set A, then we say x doesn't belongs to set A and is denoted as x∉A.

Consider set D={a,e,i,o,u}
Here,  o belongs to set D, o∈D
j doesn't belongs to set D, j∉D
Similarly, i∈D, b∉D, z∉D, p∉D, a∈D