Questions: Relational Calculus (Tuple and Domain)

Question 1

Given the relation instructor(ID, name, dept_name, salary), which tuple relational calculus expression returns the IDs of instructors whose salary is greater than 80000?

• A. {t | t in instructor and t[salary] > 80000}
• B. {t | exists s in instructor (t[ID] = s[ID] and s[salary] > 80000)}
• C. {t | for all s in instructor (s[salary] > 80000)}
• D. {t | not (t in instructor) and t[salary] > 80000}

Answer: (B)

Option B builds result tuples t that have only an ID attribute, copied from some instructor tuple s with salary over 80000. Option A returns whole instructor tuples, not just IDs. Option C requires every instructor to qualify. Option D is unsafe and not what is asked.

Question 2

What does Codd's theorem state about relational algebra and relational calculus?

• A. Relational algebra can express strictly more queries than relational calculus
• B. Relational calculus can express strictly more queries than relational algebra
• C. Relational algebra and safe relational calculus have the same expressive power
• D. Neither language can express joins

Answer: (C)

Codd showed that every relational algebra expression has an equivalent safe calculus expression and the reverse. A query language that can express all of these queries is called relationally complete.

Question 3

The tuple relational calculus expression {t | not (t in instructor)} is a safe query.

• T. True
• F. False

Answer: False

It asks for every possible tuple that is not in instructor, which is an infinite set built from values that need not appear anywhere in the database. Safe expressions restrict results to values drawn from the database and the query itself, so they always produce finite answers.

Question 4

Explain in one or two sentences how the tuple relational calculus and the domain relational calculus differ.

Model answer: In the tuple relational calculus, variables range over whole tuples, as in {t | t in instructor and t[salary] > 80000}. In the domain relational calculus, variables range over individual attribute values, as in {<i, n, d, s> | <i, n, d, s> in instructor and s > 80000}.

Both are first-order logic over relations and have the same expressive power when restricted to safe expressions. Query-by-Example (QBE) was based on the domain calculus, while SQL draws on ideas from the tuple calculus.