all students of Class X in a school. Let f : A → N be function defined by f ( x ) = roll number of the student x . Show that f is one-one but not onto. Solution No two different students of the class can have same roll number.
Therefore, f must be one-one. We can assume without any loss of generality that roll numbers of students are from to . This implies that in N is not roll number of any student of the class, so that can not be image of any element of X under f . Hence, f is not onto.
Example Show that the function f : N → N , given by f ( x ) = x , is one-one but not onto. Solution The function f is one-one, for f ( x ) = f ( x ) ⇒ x = x ⇒ x = x . Further, f is not onto, as for ∈ N , there does not exist any x in N such that f ( x ) = x = . Fig .
(i) to (iv) Example Prove that the function f : R → R , given by f ( x ) = x , is one-one and onto. Solution f is one-one, as f ( x ) = f ( x ) ⇒ x = x ⇒ x = x . Also, given any real number y in R, there exists y in R such that f ( y ) = . ( y ) = y .
Hence, f is onto. Fig . Example Show that the function f : N → N , given by f ( ) = f ( ) = and f ( x ) = x – , for every x > , is onto but not one-one. Solution f is not one-one, as f ( ) =