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I am asked to write out a Cayley table of a binary operation $\ast$ on the set $$ S = \{1, 2, 3\}$$ for which there is no solution for $\ast$ in $S$ to the equation $1 \ast x = 2$.

Here is my attempt. Is this correct? If $1 \ast x = 2$ has no solution in $S$, should that entry in the Cayley table be empty or can it be some other integer outside of $S$, i.e. 4,5, or 6?

\begin{array}{cc|c|c|c|} * && 1 & 2 & 3\\ &&&&\\ \hline 1 && & & \\ \hline 2 && 1 & 3 & 2\\ \hline 3 && 2 & 1 & 3\\ \hline \end{array}

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"No solution to $1*x=2$ means that $2$ can not appear in the first row of your Cayley table; if $2$ appeared, that would mean that $1$ star something is equal to $2$.

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    $\begingroup$ So does that mean the first row could be filled with just 1's or 3's? $\endgroup$ – mathamphetamines Sep 8 '15 at 4:21
  • $\begingroup$ @mathamphetamines Yes, that means we can only see $1$'s or $3$'s in the first row, provided $S$ is closed under the operation of $*$. The wording makes it seem as if this is the case. $\endgroup$ – Ben Sheller Sep 8 '15 at 4:22

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