Module 1
:
Real Numbers, Functions and Sequences
Lecture 1
:
Real Numbers, Functions [ Section 1.2 : Functions ]
Practice Exercises : Functions
(1)
Let
be any set. For a subset
of
, the
indicator function
of
, denoted by
, is the function
defined by
.
Prove the following statements for
:
(i)
.
(ii)
.
(iii)
.
(2)
Let
be defined by
.
Show that
is a one-one function. Is it onto also?
( This shows that the set
x
has as many points as
has ! )
(3)
Let
. For any set
, let
.
The set
is called the
preimage
of
.
Prove the following: For
,
(a)
.
(b)
.
(c)
.
(d)
is one one if and only if for every
,
is at most a singleton set.
(e)
If
, then
need not be onto.
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be any set. For a subset 
of 
, is the function 
defined by 
. 
: 
.
. 
. 
be defined by
. 
is a one-one function. Is it onto also?
. For any set 
, let 
. 
is called the preimage of 
. 
, 
. 
.
. 
, 
is at most a singleton set.
, then 
