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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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    $7\times 7=49=3\cdot 16 +1$ so what about $\psi\circ\psi$?2012-02-13
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    Note also that injectivity and surjectivity are equivalent for functions between finite sets of the same cardinality.2012-02-13
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    I haven't tried nothing, cause I don't know how to proceed. I was able to do inj. and surj. check calculating every single elements. But in this case I can't do that. @davide: what do you mean? Why $7\times 7$?2012-02-13
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    We have $\bar 7 \cdot\bar 7=\bar 1$. Thanks to that you can say what $\psi\circ\psi$ is.2012-02-13
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    Let me check if I really understand. I want to resolve the exercise just using the property of $\overline{7}$, so $x=\overline{7}$ and as a consequence $7*7=49\quad 49=3*16+1=1$ in $\mathbb{Z}_{16}$ and?2012-02-13
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    Side question: what does the bar on top of the $7$ mean?2012-02-13
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    @J.D., the bar means class mod 16.2012-02-13
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    Is my comment correct?2012-02-14

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