PostgreSQL:
SELECT name, "authorization" from card_batch_items
where card_batch_id = 63 and status = 'Completed' and not "authorization" in
(select number from transactions where date ='2007-9-25');
Simple query, more or less. Fields like so:
card_batch_items.name varchar(25) not null, card_batch_items.authorization varchar(25), # Will not be null if status = "Completed" transactions.date date not null, transactions.number integer
Simple? Not quite. If transactions.number is null for a matching row, column NOT IN (subquery) fails, because NULL = anything is NULL, and being not-true, returns nothing.