# Section 2.2: Problem 23 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let
be a structure and
a one-to-one function with
. Show that there is a unique structure
such that
is an isomorphism of
onto
.

This is similar to the previous exercise. Let
, and we define the structure
having the universe
. Note that
and is bijective. For any
, let
be the point
such that
. For any predicate symbol
, let
iff
. Similarly, for any function symbol
, let
. For any constant symbol
, let
. Then, for any predicate parameter,
iff
iff
. For any constant parameter,
. And for every function parameter,
. Therefore,
is an isomorphism of
onto
.