What is a Set?
A set is a well-defined collection of distinct objects.
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,
- So in case of the above Set
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
- Set- Builder Form or Rule Method or Description Method
- In this method, we list the properties satisfied by all elements of the set
Some examples of Roster Form vs Set-builder Form
Roster Form | Set-builder Form | |
Sets of 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
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:
- Examples:
Non Empty Set
- A set which has at least one element is called a non-empty set
- Example:
- Example:
Singleton Set
- A set containing exactly one element is called a singleton set
- Example:
- Example:
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
- Example
- 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:
, Therefore
therefore
- Example:
- 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.
- i.e.
- Empty set is a subset of every set
- i.e.
- 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
- i.e. if
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.
, 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 .
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 both
If there is a set
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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