I need some help to understand the following question:
Given that A and B are L-structures (where we take L to be a language with countably many nonlogical symbols)
such that B is an elementary substructure of A and |A| |B| (where |A| and |B| are the underlying sets of A and B, respectively) ...
... show that |B| is not definable in |A| (and is not even definable allowing points from B).
These notes http://www.math.caltech.edu/~2010-11...at6cNotes7.pdf actually helped me understand better what's going on and how one would go about proving this for specific structures, but given the above definition of A and B, I do not really know where to start.
One of hints that is given is to assume, towards a contradiction, that |B| is a A-definable subset of |A|, by some formula whose only free variable is x. Then using this formula I need to find a sentence of L true in A but false in B.
Any help would really be appreciated.