AP EAMCET · Maths · Functions
How many bijections \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) are there such that \(f(x+y)=f(x)+f(y)\) for all \(x, y \in \mathbf{Z}\) ?
- A One
- B Two
- C Three
- D Infinitely many
Answer & Solution
Correct Answer
(D) Infinitely many
Step-by-step Solution
Detailed explanation
\(f: \mathrm{Z} \rightarrow \mathbf{Z}\) \(\begin{aligned} & f(x+y)=f(x)+f(y) ; x, y \in \mathbf{Z} \\ & \therefore \quad f(x)=k x \\ \end{aligned}\) So, there are infinitely many bijections
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