# Section 2.2: Problem 24 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 an isomorphic embedding of
into
. Show that there is a structure
isomorphic to
such that
is a substructure of
.

*Suggestion*: Let be a one-to-one function with domain such that for . Form such that is an isomorphism of onto .*Remark*: The result stated in this exercise should not seem surprising. On the contrary, it is one of those statements that is obvious until you have to prove it. It says that if you can embed isomorphically into , then for all practical purposes you can pretend is a substructure of .

First we construct a set
by adding to
one element for each point of
not in
. For example, consider
such that
Then
. Note, that
is bijective, and
. Let
. Note, that by definition, for
,
, hence,
.

Now, we construct a structure
with the universe
such that
is a substructure of
, and
is isomorphic to
.

For any predicate symbol
, we define
iff
. For a constant symbol
, let
. Finally, for a function symbol
, we define
. Then, for any predicate parameter,
iff
iff
. For a constant parameter,
. And for a function parameter,
. Therefore,
is an isomorphism of
onto
.

It remains to show that
is a substructure of
. In what we did so far, we did not use the fact that
is an isomorphism of
into
, we only used the fact that it is bijective. To show that
is a substructure of
, according to (a) and (b) on page 95, we need to show the following.

(a)
is the restriction of
to
. Indeed,
iff
iff
iff
.

(b)
is the restriction of
to
(including constants). Indeed,
and
.