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I know how to determine injectivity and surjectivity for maps between regular sets, but in this case I've got some problems. How can I solve this?

Given the following map $\psi:\overline{x} \in \mathbb{Z}_{16}\mapsto \overline{7}\overline{x}\in\mathbb{Z}_{16}$. Without calculating a single element's image, and just using the properties of $\overline{7}$ in $\mathbb{Z}_{16}$, decide if $\psi$ is injective, surjective or both. If possible, find the inverse of $\psi$.

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    Is my comment correct?2012-02-14

2 Answers 2

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You may want to attack the more general question of what you can say about $\psi_a:x\mapsto ax$. Clearly, $\psi_1 = id$ and $\psi_a \circ \psi_b = \psi_{ab}$. In particular, if $ab \equiv 1$ then $\psi_a$ and $\psi_b$ are inverses of each other.

Now, consider whether there is a $b$ such that $7b\equiv 1 \bmod 16$.

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Since $7$ is relatively prime to $16$, it is a unit in the ring $\mathbb{Z}_{16}$, so $7x=7y$ implies $x=y$. Thus, multiplication by $7$ is injective. Since $\mathbb{Z}_{16}$ is a finite set, multiplication by $7$ is also bijective. The inverse of the map is also multiplication by $7$ since $7$ is its own inverse mod $16$.