Introduction

The Natural Join (symbolized by ⋈) is a specialized join operation in relational algebra that automatically combines two relations based on attributes with the same name. It simplifies query construction by eliminating the need to specify join conditions explicitly.

Syntax

R ⋈ S

Here, R and S are two relations. The Natural Join identifies and joins on attributes with matching names.

Key Features

• Automatic Join Condition: Joins on attributes with the same name.
• Duplicate Elimination: Retains only one copy of the joined attribute in the result.

Example: Basic Natural Join

Relation Employee (emp_id, name):
emp_id | name
101    | Alice
102    | Bob
103    | Carol

Relation Dependent (emp_id, dep_name):
emp_id | dep_name
101    | Charlie
102    | Dave
102    | Eve

Natural Join Result:
emp_id | name  | dep_name
101    | Alice | Charlie
102    | Bob   | Dave
102    | Bob   | Eve
            

The Natural Join automatically matches on the emp_id attribute and eliminates duplicates.

Comparison with Cartesian Product and Theta Join

• Cartesian Product: Produces all possible combinations, requiring additional filtering to make sense of the results.
• Theta Join: Allows explicit join conditions, providing flexibility but requiring more effort to construct.

Advanced Example: Multiple Common Attributes

A Natural Join can work with multiple common attributes:

Relation R (a, b, c):
a | b | c
1 | A | X
2 | B | Y

Relation S (a, b, d):
a | b | d
1 | A | P
2 | B | Q

Natural Join Result:
a | b | c | d
1 | A | X | P
2 | B | Y | Q
            

Using Renaming for Non-Matching Attributes

If the attributes to be matched have different names, use renaming to enable Natural Join:

Relation R (a, b, c):
a | b | c
1 | X | A
2 | Y | B

Relation S (e, f):
e | f
1 | P
2 | Q

Renaming and Natural Join:
ρ(S.e → a)(S) ⋈ R

Result:
a | b | c | f
1 | X | A | P
2 | Y | B | Q
            

Properties of Natural Join

• Number of Attributes: If R has K attributes, S has L attributes, and m attributes are common, the result will have K + L - m attributes.
• Number of Tuples: Depends on how many tuples satisfy the join condition.

Conclusion

• The Natural Join simplifies queries by automatically matching common attributes and removing duplicates.
• It is especially useful in normalized databases linked by foreign keys.
• When no common attributes exist, it behaves like a Cartesian Product.