Full join means include all rows from both tables. Cross join means include all combinations of rows from both tables.
A full join has a key relationship between the tables. For example:
full outer join dbo.Orders
on Customers.CustomerID = Orders.CustomerID
This will return all customers, regardless of whether they have any records in the Orders table, and all orders, regardless of whether they have a customer ID in them. There is a relationship between the tables based on the customer ID.
A cross join, on the other hand, gives you all possible combinations of the rows, without regard to a relationship between the tables.
Thus, a cross join between cities and groups give you all cities and all groups in all possible combinations.
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