Q. 2

Define each of the following:

(i) injective function

(ii) surjective function

(iii) bijective function

(iv) many - one function

(v) into function

Give an example of each type of functions.

Answer :

1)injective function


Definition: A function f: A B is said to be a one - one function or injective mapping if different elements of A have different f images in B.


A function f is injective if and only if whenever f(x) = f(y), x = y.


Example: f(x) = x + 9 from the set of real number R to R is an injective function. When x = 3,then :f(x) = 12,when f(y) = 8,the value of y can only be 3,so x = y.




(ii) surjective function


Definition: If the function f:AB is such that each element in B (co - domain) is the ‘f’ image of atleast one element in A , then we say that f is a function of A ‘onto’ B .Thus f: AB is surjective if, for all bB, there are some aA such that f(a) = b.


Example: The function f(x) = 2x from the set of natural numbers N to the set of non negative even numbers is a surjective function.


(iii) bijective function


Definition: A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y.Alternatively, f is bijective if it is a one - to - one correspondence between those sets, in other words, both injective and surjective.


Example: If f(x) = x2,from the set of positive real numbers to positive real numbers is both injective and surjective.Thus it is a bijective function.


(iv)many - one function


Defintion : A function f: AB is said to be a many one functions if two or more elements of A have the same f image in B.


trigonometric functions such as sinx are many - to - one since sinx = sin(2 + x) = sin(4 + x) and so one…


(v) into function


Definition: If f:AB is such that there exists atleast one element in co - domain , which is not the image of any element in the domain , then f(x) is into.


Let f(x) = y = x – 1000


x = y + 1000 = g(y) (say)


Here g(y) is defined for each y I , but g(y) N for y ≤ − 1000. Hence,f is into.


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