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Question
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Solution
f : Z → Z, defined by f(x) = x − 5
Injection test :
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
f(x) = f(y)
x - 5 = y - 5
x = y
So, f is 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
x - 5 = y
x = y + 5, which is in Z.
So, f is a surjection and f is a bijection
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