Give an example of a function: Which is not one-one but onto. - Vidyadip

Question

Give an example of a function:
Which is not one-one but onto.

✓

Answer

Which is not one-one but onto.
$\text{f}:\text{Z}\rightarrow\text{N}\cup\{0\}$ given by f(x) = |x|
Infectivity: Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
Implies that |x| = |y|
Implies that $\text{x}=\pm\text{y}$
Therefore, different elements of domain f may give the same image.
Therefore, f is not one-one.
Subjectivity: 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
Implies that |x| = y
Implies that $\text{x}=\pm\text{y},$
Which is an element in Z (domain).
Therefore, for every element in the co-domain, there exists a pre-image in the domain.
Thus, f is onto.

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