This is similar to the previous exercise. Let $|\mathfrak{B}|=\mbox{ran}g$ , and we define the structure $\mathfrak{B}$ having the universe $|\mathfrak{B}|$ . Note that $g:|\mathfrak{A}|\rightarrow|\mathfrak{B}|$ and is bijective. For any $b\in|\mathfrak{B}|$ , let $h(b)$ be the point $a\in|\mathfrak{A}|$ such that $g(a)=b$ . For any predicate symbol $P$ , let $<b_{1},\ldots,b_{n}>\in P^{\mathfrak{B}}$ iff $<h(b_{1}),\ldots,h(b_{n})>\in P^{\mathfrak{A}}$ . Similarly, for any function symbol $f$ , let $f^{\mathfrak{B}}(b_{1},\ldots,b_{n})=g(f^{\mathfrak{A}}(h(b_{1}),\ldots,h(b_{n})))$ . For any constant symbol $c$ , let $c^{\mathfrak{B}}=g(c^{\mathfrak{A}})$ . Then, for any predicate parameter, $<a_{1},\ldots,a_{n}>\in P^{\mathfrak{A}}$ iff $<h(g(a_{1})),\ldots,h(g(a_{n}))>\in P^{\mathfrak{A}}$ iff $<g(a_{1}),\ldots,g(a_{n})>\in P^{\mathfrak{B}}$ . For any constant parameter, $g(c^{\mathfrak{A}})=c^{\mathfrak{B}}$ . And for every function parameter, $g(f^{\mathfrak{A}}(a_{1},\ldots,a_{n}))=g(f^{\mathfrak{A}}(h(g(a_{1})),\ldots,h(g(a_{n}))))=f^{\mathfrak{B}}(g(a_{1}),\ldots,g(a_{n}))$ . Therefore, $g$ is an isomorphism of $\mathfrak{A}$ onto $\mathfrak{B}$ .