This sounds like an "Unmatched" query, for which (I think) there is a query wizard. If you enabled wizards in your DB, you could try to make a new query that does a JOIN via the query wizards. It would show you if there is an unmatched record. However, whichever table is the "reference" table SHOULD have an index on that serial-number field. Not required, but makes it faster.
It MIGHT look something like this:
SELECT A.SerialNum, B.SerialNum, IIF( IsNull(B.SerialNum), "No", "Yes" )
FROM A LEFT JOIN B ON A.SerialNum = B.SerialNum ;
Not saying this is exactly what you want. Try the query wizard or search on "Unmatched Query" to see other examples of same.