Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
f : Z → Z, defined by f(x) = x - 5

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Answer

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

Therefore, 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.

Therefore, f is a surjection and f is a bijection.

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