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### Problem 5

Give a careful proof for a skeptic that the property "For each $c \in S$, the equation $x*x=c$ has a solution $x$ in $S$." is a structural property.

**Solution**

Let $\langle S,* \rangle$ and $\langle S',*' \rangle$ be binary algebraic structures and let $\phi : S \rightarrow S'$ be an isomorphism. $S$ has the property that for each $c \in S, \; \exists \; x \in S$ s.t. $x*x=c$. Let $c' \in S'$ and let $c \in S$ s.t. $\phi(c)=c'$. Find $x \in S$ s.t. $x*x=c$. Then $\phi(x*x)=\phi(c)=c'$, so $\phi(x)*'\phi(x)=c'$. If we denote $\phi(x)$ by $x'$, then we see that $x'*'x'=c'$ which means $x'*'x'=c'$ has a solution $x'$ in $S'$ for each $c' \in S'$. So the property "For each $c \in S'$, the equation $x*x=c$ has a solution $x$ in $S$" is a structural property.