In relational algebra, the natural join of two tables can be described as:

Concept

A join combines tuples from two relations under a matching condition. A natural join uses equality on every same-named attribute and returns one copy of each common attribute in the result schema.

Using only basic relational-algebra operators, this requires filtering the Cartesian product and then projecting the union of the attribute names: R ⋈ S = π_attrs(R)∪attrs(S)(σ_common-equality(R × S)).

Application

1. Cartesian product: form R × S, which contains every tuple pair and both relations’ attribute copies.

2. Filter: retain only pairs whose values agree on every shared attribute name. This intermediate relation is the filtered Cartesian product.

3. Project: select the combined attribute list and retain a single copy of each shared attribute, producing the natural-join schema.

Cross-check and contrast

• A plain Cartesian product performs no equality filtering and keeps both attribute copies.

• Union appends tuples from union-compatible relations; it does not construct matching tuple pairs.

• Adding selection to an already filtered Cartesian product repeats the filtering role and still leaves the duplicate shared-attribute copies.

• Adding projection to the filtered Cartesian product supplies the remaining schema step: retaining one copy of each common attribute.

• A classroom formula such as σ(R × S) describes the equality-filtering stage, or an equijoin if both common-attribute copies remain. A natural join also applies the schema projection so that each common attribute appears once.

Result: Combination of Projection and filtered Cartesian Product.