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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