Let $J_5=\{0,1,2,3,4\}$. Then $J_5-\{0\}=\{1,2,3,4\}$. A student tries to define a function $R:J_5-\{0\}\to J_5-\{0\}$ as follows: For each $x\in J_5-\{0\}$,
$R(x)$ is the number $y$ so that $(xy)\bmod 5=1$.
Student B claims that $R$ is not so well defined. Who is right?
My answer. Consider $R(0)$, where $(0\cdot y)\bmod 5=1$. There is no corresponding value of $y$ for $x=0$, and therefore not all elements in the domain have an image in the co-domain, and therefore it is not well defined.
However my answer said "Consider $R(3). $Then $y$ can be $2$ or $7$. Since an element of the domain has $2$ images, the function is not well defined.
I agree with what my asnswer said, but will my attempt get me the marks too?
My question would be "Will a function be considered a function if an element in the domain have an image in the co-domain?"