Relational Algebra Operations

Relations are sets so we can apply set operators. However, we want the results to be relations.

The two operands $R$ and $S$ must be type-compatible (except for CROSS PROD.).

Same number of attributes.
Same domain for attributes.

Properties of operations:

Both UNION (Relational Operation) and INTERSECTION (Relational Operation) are commutative: $R \cup S = S \cup R$
They can also both be treated as $n$ -ary operations applicable to any number of relations as both are associative operations: $R \cup (S \cup T) = R \cup S \cup T$

UNION

UNION (Relational Operation)
The binary operation UNION, denoted by $\cup$ , creates a relation that includes all tuples that includes all tuples from either operand.

The result of $R \cup S$ is a relation that includes all tuples that are either in $R$ , in $S$ , or in both. Duplicate tuples are eliminated.

Examples:

Retrieve the SSNs of all employees who work in dept 5. or directly supervise an employee who works in dept 5. $D E P 5_{E M P S} R E S U L T_{1} R E S U L T_{2} (S S N) R E S U L T \leftarrow σ_{D N O = 5} (E M P L O Y E E) \leftarrow π_{S S N} (D E P 5_{E M P S}) \leftarrow π_{S U P E R S S N} (D E P 5_{E M P S}) \leftarrow R E S U L T_{1} \cup R E S U L T_{2}$

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INTERSECTION

INTERSECTION (Relational Operation)
The binary operation INTERSECTION, denoted by $\cap$ , creates a relation that includes all tuples that are in both operands.

The result of $R \cap S$ is a relation that includes all tuples that are both in $R$ and $S$ .
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DIFFERENCE

DIFFERENCE (Relational Operation)
The binary operation SET DIFFERENCE (also called MINUS or EXCEPT), denoted by $-$ , creates a relation that includes all the tuples in the first operand that are not in the second operand.

The result of $R - S$ is a relation that includes all tuples that are in $R$ but not $S$ .
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CARTESIAN PRODUCT

CARTESIAN PRODUCT (Relational Operation)
The binary operation CARTESIAN PRODUCT (or CROSS PRODUCT), denoted by $\times$ , creates a relation that combines tuples from two operands in a combinatorial fashion.

The result of $R (A_{1}, A_{2}, \dots, A_{n}) \times S (B_{1}, B_{2}, \dots, B_{m})$ is a relation with degree $n$ + $m$ attributes: $Q (A_{1}, A_{2}, \dots, A_{n}, S (B_{1}, B_{2}, \dots, B_{m})$ . The resulting relation state has one tuple for each combination of tuples - one from $R$ and one from $S$ .

Operands do not have to be type-compatible.

Generally, cross product is not meaningful but it can precede other operations. To keep only combinations where the DEPARTMENT is related to the EMPLOYEE, we can add a SELECT operation:
$F E M A L E_{E M P S} E M P N A M E S E M P_{D E P E N D E N T S} A C T U A L_D E P S R E S U L T \leftarrow σ_{S E X =^{'} F^{'}} (E M P L O Y E E) \leftarrow π_{FNAME, LNAME, SSN} (F E M A L E_{E M P S}) \leftarrow E M P N A M E S \times D E P E N D E N T \leftarrow σ_{S S N = E S S N} (E M P_{D E P E N D E N T S}) \leftarrow π_{F N A M E, L N A M E, D E P E N D E N T_{N} A M E} (A C T U A L_D E P S)$ Link to original

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Relational Algebra Operations

UNION

UNION (Relational Operation)

INTERSECTION

INTERSECTION (Relational Operation)

DIFFERENCE

DIFFERENCE (Relational Operation)

CARTESIAN PRODUCT

CARTESIAN PRODUCT (Relational Operation)

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