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 .