# Show that f : R <

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 given by f(x) = x [x]

To Prove: – f(x) = x [x], is neither one – one nor onto

Check for Injectivity:

Let x be element belongs to Z i.e such that

So, from definition

f(x) = x [x]

f(x) = 0 for Therefore,

Range of f = [0,1] ≠ R

Hence f is not One – One function

Check for Surjectivity:

Since Range of f = [0,1] ≠ R

Hence, f is not Onto function.

Thus, it is neither One – One nor Onto function

Hence Proved

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