• ### Object : Everything in this universe whether living or non living is called an object.

• Set : –  A set is a collection of well-defined objects.
• ### A set is a collection of things that have something in common or following a rule.

• Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.
• Elements of Set:
• The things or objects  in the set are called “ elements “
•  Notation of a Set:  A set is usually denoted by capital letters and elements are denoted by small letters

### The pair of curly braces {  } denotes a set.

• The elements of set are written inside a pair of curly braces separated by commas.
• The elements of a set may be written in any order.
• The elements of a set must not be repeated.
• The Greek letter Epsilon ‘∈’ is used for the words ‘belongs to’, ‘is an element of’, etc.
• Therefore, x ∈ A will be read as ‘x belongs to set A’ or ‘x is an element of the set A’.
• The symbol ‘∉’ stands for ‘does not belongs to’ also for ‘is not an element of’.
• Ex: –  The collection of positive numbers less than 10’ is a set.
• The collection of good students in your class’ is not a set.
• The collection of first five months of a year is a set.
• The collection of rich man in your town is not a set.
• The collection of vowels in English alphabets. This set contains five elements,namely a, e, i, o, u.

• ## For example:

• ### Therefore, N = {1, 2, 3, 4, 5 }

• Empty Set or Null Set: A set which does not contain any element is called an empty set, or the null set or the void set         and it is denoted by and is read as phi.

In roster form, is denoted by {  }.

An empty set is a finite set,

For example: (a) The set of whole numbers less than 0.
Note: ≠ {0}

The cardinal number of an empty set, i.e., n() = 0

•     ∅  has no element.
{0} is a set which has one element 0.

• Singleton Set:  A set which contains only one element is called a singleton set.
Ex A = {x : x is neither prime nor composite}

B = {x : x is a whole number, x < 1}         is a singleton set containing one element, i.e., 1.

•  This set contains only one element 0 and is a singleton set.
Let A = {x : x  N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is 4.
Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.

• UNIVERSAL SET:

A universal set is a set  which contains all objects including itself.

All sets under consideration will be subsets of a fixed set. We call this set the Universal Set.

A set which contains all the elements of other given sets is called a universal set.

The symbol for denoting a universal set is   or ξ.

•
• Subset:- If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B, and we write it as A   B or B  A

If all elements of set A are present in B , then A is said to be subset of B denoted  by  A  B

The symbol   stands for ‘is a subset of’ or ‘is contained in’
Every set is a subset of itself,
i.e., A
A, B B.
Empty set is a subset of every set.
Symbol ‘’ is used to denote ‘is a subset of’ or ‘is contained in’.
A B  means A is a subset of B or A is contained in B.
B A means B contains A.

• For example;

1.     Let A = {2, 4, 6}  B = {6, 4, 8, 2}    Here A is a subset
of B

2.   Notes: If A B and B A, then A = B, i.e., they are equal sets.

• Super Set: Let A and B are two sets. If  A B , then B is called the super set of A.

Whenever a set A is a subset of set B, we say that  B is a superset of A and we write, B A.
Symbol
is used to denote ‘ is a super set of

For example;

A = {a, e, i, o, u}              B = {a, b, c, …………., z}
Here A
B i.e., A is a subset of B but B A i.e., B is a super set of A