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
⇒ 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
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
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