Q. 145.0( 1 Vote )

# If A = {1, 2, 3}, show that a one – one function f : A → A must be onto.

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

a≠b

f(a)≠f(b) for all

f(a) = f(b)

a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Now, f: A A where A = {1, 2, 3} and its a One – One function

To Prove: – A is Onto function

Since it is given that f is a One – One function,

Three elements of A = {1, 2, 3} must be taken to 3 different elements of co – domain A = {1, 2, 3} under f.

Thus by definition of Onto Function

f has to be Onto function.

Hence Proved

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