I don't understand what , and are. Suppose defines and let . The required formula must have three free variables and no constants, except possibly those that occur in .
We have , i.e., . Therefore, is defined by where .
The set of all -tuples for which is called a definable subset of The set is the universe of structure
Sometimes I think I do not understand what a definable subset is.
Here is an exercise on definable subset:
Let be the underlying set for structure Suppose that is a definable subset of
Suppose we rearrange the order of the -tuples. Consider the set of all such that is in Show that this set is definable.
Then I say that the the sought formula is as follows:
where and defines
While it may be correct, I feel that there is a simpler formula. What do you think?