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:A→B 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: A→B is surjective if, for all b∈B, there are some a∈A 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) = x^{2},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: A→B 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:A→B 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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