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Joins in Relational Calculus
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Now let s take a look at joins from a calculus perspective. Remember, calculus expressions tell us what we would like the resultant table to look like as opposed to how we get it. Let s start with the Cartesian product: we want a set of rows made up of combinations of rows from each of the contributing tables. Figure 3-4 shows how we can envisage this. We are looking at two tables, so we need two fingers to keep track of the rows. Finger m looks at each row of the Member table in turn. Currently it is pointing at row 3. For each row in the Member table, finger t will point to each row in the Type table.
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Member table
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Type table
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Figure 3-4. Row variables m and t point to each row of their respective tables.
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Listing 3-5 shows the relational calculus expression.
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Listing 3-5. Relational Calculus for Cartesian Product {m, t | Member(m), Type(t)}
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Listing 3-6 shows the SQL that is very similar to this relational calculus expression.
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Listing 3-6. Alternative SQL for a Cartesian Product to Produce Table in Figure 3-3a SELECT * FROM Member m , Type t
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The SQL statement in Listing 3-6 is equivalent to the one in Listing 3-3 that used the CROSS JOIN keyword. They will both return the same set of rows. Which you use doesn t matter in fact, in some products when you type Listing 3-6, it will automatically replace it with Listing 3-3. Other products do not implement the key phrase CROSS JOIN, so you have to use the alternative statement.
CHAPTER 3 A FIRST LOOK AT JOIN S
Now let s consider the join. As the two fingers traverse the two tables, they cover every combination of rows. For the join we have the extra condition that we want to retrieve rows only where the membership type from each table is the same. The pair of rows depicted in Figure 3-4 satisfies that condition and so will be retrieved. If m stays where it is and t moves, then the condition will no longer be satisfied. We can express this in calculus notation as in Listing 3-7 where we have just added the extra condition to the right side.
Listing 3-7. Relational Calculus for Join {m, t | Member(m), Type(t) and m.MemberType = t.Type}
We can translate the calculus expression directly into an SQL statement as in Listing 3-8. The extra condition is represented by a WHERE clause.
Listing 3-8. Alternative SQL for a Join to Produce the Table in Figure 3-3b SELECT * FROM Member m , Type t WHERE m.MemberType = t.Type
The SQL statement in Listing 3-8 is based on relational calculus in that it says what the rows to be retrieved are like. We want combinations of rows from Member and Type where the membership types are the same. The statement is equivalent to the statement in Listing 3-4, which uses the INNER JOIN key phrase. Once again, which one you use does not matter it just depends how you find yourself thinking about the query. Sometimes there is a possibility that the way you express the query may affect the performance, and we will talk about this more in 9. Actually, most database products are pretty smart at optimizing or finding the quickest way to perform a query regardless of how you express it. For example, in SQL Server, the queries in Listings 3-4 and 3-8 are carried out in the same way. In fact, in SQL Server, if you type the code in Listing 3-8 into the default interface, that code will be replaced by Listing 3-6. Even though the expression in Listing 3-8 is not based directly on the algebra, we can see how the algebra is reflected. The second line is the Cartesian product, and the last line is a select and there we have our algebra definition of a join.
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