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Monday, 26 October 2015

Set, Relation & Function ,type of sets,

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 { }.
  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 anull 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.
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
A = {4, 5, 6} and
C = {6, 5, 4} are equal, i.e. A = C.
 function and relation,notation of function, domain and range,special function,composition of functions,injection
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 function and relation,notation of function, domain and range,special function,composition of functions,injection
set part2.................................
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