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Question
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
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Solution
f : Z → Z, given by f(x) = x2
Injection test :
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
f(x) = f(y)
x2=y2
x = ±y
So, f is not an injection .
Surjection test:
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
x2= y
x=± `sqrty ` which may not be in Z.
For example, if y = 3,
x = ± `sqrt3 ` is not in Z.
So, f is not a surjection.
So, f is not a bijection.
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