Database Transaction Integrity and Relational Algebra Operations

The serializability problem arises when transactions interfere, potentially leading to incorrect results even if individual transactions succeed. If a transaction involves a commit, it is serializable; otherwise, it may not be. This graph technique is applicable only when multiple modules execute transactions simultaneously.

A successful transaction can yield incorrect results due to interference with other transactions. Three common problems include:

1. Lost Update: Occurs when two transactions read the same data, and one transaction's update is lost because the other transaction writes its update later without considering the first.
2. Unconfirmed Unit: Happens when a transaction retrieves or updates a tuple that was modified by another transaction that has not yet completed.
3. Inconsistent Analysis: Arises when two transactions operate on the same data, and one commits before the other, leading to an inconsistent view.

Definition of a Transaction: A sequence of operations triggered by input messages, generating output, and fulfilling these conditions:

1. Atomicity: A transaction either completes entirely or is canceled and annulled.
2. Consistency: Ensures data integrity at the beginning and end of the transaction.
3. Durability: Confirmed actions are permanent and not lost.
4. Isolation: For any pair of transactions, the execution ensures that one transaction finishes before the other begins its effects.

Relational Algebra Operations:

Which relational algebra operators can perform the following operations?

• a) Intersection (R ∩ S) = (R ∪ S) - ((R - S) ∪ (S - R))
• b) Cartesian product, preceded and followed by SELECT and PROJECT operations.

Set Operations in Relational Algebra:

• a) Intersection (R ∩ S) is equivalent to (R ∪ S) - ((R - S) ∪ (S - R)).
• b) Selecting and projecting attributes involves operators that work on the same attribute domain, similar to a JOIN operation.

The JOIN Operator in Relational Algebra:

1. Definition: Joins two relations, R1 and R2, which do not necessarily have compatible schemas, but share at least one common attribute with a qualifying domain. The resulting relation, R3, combines the schemas of R1 and R2. Its tuples are formed from the Cartesian product R1 x R2 that satisfy a specified qualification Q.
2. Example: R3 = R1 ⋈_Q R2
3. Types of JOIN:
   • Equijoin: The qualification Q uses the equality operator (=).
   • Theta Join: The qualification Q uses any comparison operator (!=, <, >, etc.).
   • Natural Join: An equijoin performed on all common attributes between R1 and R2.
   • Self-Join: A natural join performed on the same relation.
   • Semijoin: Produces a relation with the schema of R1, containing only those tuples from R1 that have matching tuples in R2 based on the join condition.