GATE CS 2012 Q51 — Tuple Calculus and SQL

To find the number of tuples returned by the SQL query, let us evaluate it step-by-step:

Step 1: Evaluate the Subquery

The subquery is:

SELECT B.Age 
FROM B 
WHERE B.Name = 'Arun'

Looking at relation $B$, there is no tuple where the Name attribute is 'Arun'. Therefore, this subquery returns an empty set ($\emptyset$).

Step 2: Evaluate the Outer Query Condition

The outer query filters tuples from relation $A$ using the condition:

$$\text{A.Age} > \text{ALL } (\emptyset)$$

In SQL, the > ALL operator checks if a value is greater than every value in the subquery result. Mathematically, this is a universal quantification:

$$\forall y \in S, \text{A.Age} > y$$

When the set $S$ is empty ($S = \emptyset$), this condition is vacuously true for any value of A.Age. Thus, the WHERE clause evaluates to TRUE for every tuple in relation $A$.

Step 3: Count the Resulting Tuples

Since the condition is true for all tuples in $A$, the query selects the Id of every tuple in $A$:

• For $\text{Id} = 12$, $\text{Age} = 60 \implies \text{TRUE}$
• For $\text{Id} = 15$, $\text{Age} = 24 \implies \text{TRUE}$
• For $\text{Id} = 99$, $\text{Age} = 11 \implies \text{TRUE}$

Relation $A$ contains exactly $3$ tuples. Therefore, the result of the query contains $3$ tuples.