Set, Relation & Function ,type of sets,
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October 26, 2015 |
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Set,
Relation & Function
Set is any collection
of distinct and distinguishable objects of our intuition or thought’.
Following
are the some examples of sets:
•The collection of vowels in English alphabets, i.e. A
= {a, e, i,
o, u}.
•The collection of all states in the Indian Union.
•The collection of all past presidents of the Indian Union
etc.The
following sets, we will use frequently
In
this session
and following sessions:
•N : For the set of
natural numbers
•Z or I: For the set of integers
•Z+
or I+: For the set of all positive
integers
•Q : For the set of
all rational numbers
•Q+ :
For the set of all positive rational numbers
•R : For the set of
all real numbers
•R+ :
For the set of all positive real numbers
•C : For the set of
all complex numbers
Representation of a Set
A set is often
represented in the following two ways:(I)Roster method (Tabular form)
In
this method a set is described by listing
elements separated by commas, within braces { }.
elements separated by commas, within braces { }.
For example, the set of even natural
numbers can be described as {2, 4, 6, 8, ...}.
(II)Set Builder Method
In this method, a
set is described by a characterizing property
(x) of its element x. In such a
case the set is described by {x : P(x) holds} or {x / P(x) holds}
The symbol ‘|’ or ‘:’ is read as ‘such that’.In
this representation the set of all even natural numbers can be written as : {x
/ x = 2n for }
Types of Sets
Empty sets: A
set having no element is called
an empty set. It is also known as null
set or void set. It is denoted by phi
Singleton set: A
set having single element is called singleton set. For
example, {2}, {0}, {5} are singleton set.
Finite set: A
set is called a finite set if it is
called either void set or its elements can be
counted by natural numbers and process
of listing terminates at a certain natural numbers.
of listing terminates at a certain natural numbers.
For
example, {1, 2, 4, 6} is a finite set because it has four elements.
Infinite set: A
set which is not a finite set, i.e. the counting up of whose elements is
impossible, is called an infinite set.For
example:
(i)The
set of all straight line in a given plane.
(ii)The
set of all natural numbers.
The
set of real numbers between ‘1’ and 2
Equivalent and Equal Sets
Equivalent sets:
Two finite sets A and B are equivalent if their cardinal number is same, i.e.
n(A) = n(B).
Equal sets: Two
sets A and B are said to be equal if every element of A is a member of B, and
every element of B is a member of A.
For
example:
A
= {4, 5, 6} and
B = {a, b, c} are equivalent but
B = {a, b, c} are equivalent but
A
= {4, 5, 6} and
C = {6, 5, 4} are equal, i.e. A = C.
C = {6, 5, 4} are equal, i.e. A = C.
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