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Question
Give an example of a function which is not one-one but onto ?
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Solution
which is not one-one but onto
f: Z → N ∪ {0} given by f(x) = |x|
njectivity:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
⇒ |x| = |y|
⇒ x= ± y
So, different elements of domain f may give the same image.
So, f is not one-one.
Surjectivity:
Let y be any element in the co domain (Z), such that f(x) = y for some element x in Z(domain).
f(x) = y
⇒ |x| = y
⇒ x = ± y
which is an element in Z (domain).
So, for every element in the co-domain, there exists a pre-image in the domain.
Thus, f is onto.
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