I am given this function:
$f(n) = 2n+1$ if n is even, and $n-1$ is n is odd. I am asked to prove that the image of $2\mathbb{Z}$ is equal to the inverse image of $4\mathbb{Z}$.
My approach is to prove that they are both equal to $(4\mathbb{Z} +1)$. So far I have proved that $f(2\mathbb{Z}) \subseteq (4\mathbb{Z} +1)$ and that $f^{-1}(4\mathbb{Z}) \subseteq (4\mathbb{Z} +1)$ but I couldn't prove the other two directions.
Any help please?
Proving $f(2 \mathbb{Z}) \supseteq 4 \mathbb{Z} + 1$: if I give you some integer $k$, can you find an even number $n$ such that $f(n) = 4k+1$?
Proving $f^{-1}(4 \mathbb{Z}) \supseteq 4 \mathbb{Z} + 1$: if I give you some integer $k$, can you give me an integer $m$ such that $f(4m+1) = 4k$?