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Question
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
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Solution
f : N → N, given by f(x) = x2
Injection test :
Let x and y be any two elements in the domain (N), such that f(x) = f(y).
f(x)=f(y)
x2=y2
x=y (We do not get ± because x and y are in N)
So, f is an injection .
Surjection test :
Let y be any element in the co-domain (N), such that f(x) = y for some element x in N(domain).
f(x) = y
x2= y
x =`sqrty , ` which may not be in N.
For example, if y = 3 ,
x=`sqrt 3 ` is not in N.
So, f is not a surjection.
So, f is not a bijection.
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