Q. 44.3( 3 Votes )

Let A = {–1, 0, 1} and f = {(x, x2) : x A}. Show that f : A A is neither one – one nor onto.

Answer :

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, We have, A = {–1, 0, 1} and f = {(x, x2) : x A}.


To Prove: – f : A A is neither One – One nor onto function


Check for Injectivity:


We can clearly see that


f(1) = 1


and f( – 1) = 1


Therefore


f(1) = f( – 1)


Every element of A does not have different image from A


Hence f is not One – One function


Check for Surjectivity:


Since, y = – 1 be element belongs to A


i.e in co – domain does not have any pre image in domain A.


Hence, f is not Onto function.


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