(b) Onto function (Surjection): A function f A B
: →
is said to be an onto
function if each element of B is the image of some element of A.
i.e., f A B
: → is onto if the image of f is the entire codomain, i.e. if
f A B
( ) = . i.e., f maps A onto B.
(c) One-to-one onto Function (Bijection): A function that is both one-to-one
and onto is called a “Bijection”. Such a function maps each and every element
of A to exactly one element of B, with no elements left over. Fig. below shows
bijection.
(d) Invertible function: A function f A B
: →
is invertible if its inverse
relation f
–1 is a function from B to A.
A function f A B
: →
is invertible if and only it is both one-to-one and
onto.
Ì Exam ple 0.1.15: Find whether the function f x x
( ) =
2 from the set of
integers to the set of integers is one-to-one.
Solu tion
The function f x x
( ) =
2 is not one-to-one as, for example f
f
( )
( )
1
1 1
= − = , but
1 1
≠ − .
Ì Exam ple 0.1.16: Given f is a function f A B
: → where A = {a, b, c, d}
and B = {1, 2, 3} with f (a) = 3, f (b) = 2, f (c) =1, and f(d) = 3. Is the
function f an onto function?
Introduction
13
a
b
c
d
w
x
y
z
Fig. A Bijection
a
b
c
d
e
w
x
y
z
Fig. A Surjection