__What is a Set?
__A set is a well-defined collection of distinct objects.

Example:

__What is an element of a Set?__

- The objects in a set are called its elements.
- So in case of the above Set , the elements would be and . We can say,

- Usually we denote Sets by CAPITAL LETTERs like etc. while their elements are denoted in small letters like
- If is an element of , then we say belongs to and we represent it as
- If is not an element of , then we say that does not belong to and we represent it as

__How to describe a Set?__

- Roaster Method or Tabular Form
- In this form, we just list the elements
- Example or

- Set- Builder Form or Rule Method or Description Method
- In this method, we list the properties satisfied by all elements of the set

Example

Some examples of Roster Form vs Set-builder Form

Roster Form | Set-builder Form | |

__Sets of Numbers__

- Natural Numbers

- Integers

- Whole Numbers

- Rational Numbers

__Finite Sets & Infinite Sets__

Finite Set: A set where the process of counting the elements of the set would surely come to an end is called finite set

- Example: All natural numbers less than
- All factors of the number

Infinite Set: A set that consists of uncountable number of distinct elements is called infinite set.

Example: Set containing all natural numbers

__Cardinal number of Finite Set__

The number of distinct elements contained in a finite set is called the cardinal number of and is denoted by

- Example then
- . Therefore and
- , Therefore and

__Empty Set__

- A set containing no elements at all is called an empty set or a null set or a void set.
- It is denoted by (phai)
- In roster form you write
- Also
- Examples:

__Non Empty Set__

- A set which has at least one element is called a non-empty set
- Example: or

__Singleton Set__

- A set containing exactly one element is called a singleton set
- Example: or

__Equal Sets__

- Two set and are said to be equal sets and written as if every element of is in and every element of is in
- Example and

- It is not about the number of elements. It is the elements themselves.
- If the sets are not equal, then we write as

__Equivalent Sets__

- Two finite sets and are said to be equivalent, written as , if , that is they have the same number of elements.
- Example: and , Therefore and therefore

- Note: Two equal sets are always equivalent but two equivalent sets need not be equal.

__Subsets__

- If and are two sets given in such a way that every element of is in , then we say is a subset of and we write it as
- Therefore is and then
- If is a subset of , we say is a super set of and is written as
- Every set is a subset of itself.
- i.e. , etc.

- Empty set is a subset of every set
- i.e.

- If and , then
- Similarly, if , then and
- If set contains elements, then there are subsets of

__Power Set__

- The set of all possible subsets of a set is called the power set of , denoted by . If A contains elements, then
- i.e. if , then
- Empty set is a subset of every set
- So in this case the subsets are

Proper Subset

Let be any set and let be any non-empty subset. Then is called a proper subset of , and is written as , if and only if every element of is in , and there exists at least one element in which is not there in .

i.e. if and then

Please note that has no proper subset

A set containing elements has proper subsets.

i.e. if , then the number of proper subsets is

Universal Set

If there are some sets in consideration, then there happens to be a set which is a super set of each one of the given sets. Such a set is known as universal set, to be denoted by or .

i.e. if , and , then or

Operations on Sets

Union of Sets

The union of sets and , denoted by , is the set of all those elements, each one of which is either in or in or in both and

If there is a set and , then

So if then which means or

And if which means or

Interaction of Sets

The intersection of sets and is denoted by , and is a set of all elements that are common in sets and .

- if and , then as is the only common element.

Thus then i.e. and

And if i.e. and

Disjointed Sets

Two sets and are called disjointed, if they have no element in common. Therefore:

i.e. and

Intersecting sets

Two sets are said to be intersecting or overlapping or joint sets, if they have at least one element in common.

Therefore two sets and are overlapping if and only if

Intersection of sets is Commutative

i.e. for any sets and

Intersection of sets is Associative i.e. for any sets,

, then

Since , so

For any sets and we have

and

for every set

Difference of Sets

For any two sets and , the difference is a set of all those elements of which are not in .

i.e. if and , Then and

Therefore , then then but

If then

Complement of a Set

Let be the universal set and let . The the complement of , denoted by is the set if all those elements of which are not in .

i.e. let and , then

Thus clearly and

Please note

and

and

Disruptive laws for Union and Intersection of Sets

For any three sets , , we have the following

Say and

Therefore and and hence equal

Say and

Then and and hence equal

Disruptive laws for Union and Intersection of Sets

De-Morgan’s Laws

Let and be two subsets of a universal set , then

Let and and

Then , therefore

and

Therefore . Hence proven.

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