Let there be two relations $X$ and $Y$ as shown. $X$ has three columns $P$, $Q$ and $R$. $Y$ has two columns $P$ and $S$. Table X Table Y Consider that the following tuple relational calculus expression is evaluated. $$ \{t \mid t \in X \land \exists z \in X (t[P] = z[P]) \land \exists m \in Y (m[P] = t[P] \land m[S] > 1)\} $$ The number of tuples that will be returned is __________ . (Answer in integer)P Q R P1 Q1 R1 P2 Q2 R2 P3 Q3 R2 P S P1 10 P1 15 P2 20 P3 1
The problem requires evaluating a given tuple relational calculus expression on two relations, X and Y, to determine the number of tuples returned.
Relations X and Y Data
Relation X has columns P, Q, and R. Relation Y has columns P and S.
| P | Q | R |
|---|---|---|
| P1 | Q1 | R1 |
| P2 | Q2 | R2 |
| P3 | Q3 | R2 |
| P | S |
|---|---|
| P1 | 10 |
| P1 | 15 |
| P2 | 20 |
| P3 | 1 |
Tuple Relational Calculus Expression Analysis
The given expression is:
$ \{t \mid t \in X \land \exists z \in X (t[P] = z[P]) \land \exists m \in Y (m[P] = t[P] \land m[S] > 1)\} $Let's break down the conditions:
- $t \in X$: The tuple $t$ must belong to relation X.
- $\exists z \in X (t[P] = z[P])$: There must exist a tuple $z$ in X such that its P value matches $t$'s P value. Since $t$ is already from X, this condition is always met and does not impose additional restrictions.
- $\exists m \in Y (m[P] = t[P] \land m[S] > 1)$: This is the primary filter. For a tuple $t$ from X, there must be at least one tuple $m$ in relation Y where $m$'s P value equals $t$'s P value, AND $m$'s S value is strictly greater than 1.
In essence, we are looking for tuples in X whose P values exist in Y with an associated S value greater than 1.
Evaluating Conditions for Relation X Tuples
We examine each unique P value present in relation X (which are P1, P2, P3) against the condition $\exists m \in Y (m[P] = t[P] \land m[S] > 1)$.
-
For P = P1:
In relation Y, tuples with P=P1 are (P1, 10) and (P1, 15).
Checking the S value: 10 > 1 (True) and 15 > 1 (True).
Since there's at least one tuple in Y meeting the criteria (S > 1), P1 satisfies the condition. The tuple (P1, Q1, R1) from X will be returned.
-
For P = P2:
In relation Y, the tuple with P=P2 is (P2, 20).
Checking the S value: 20 > 1 (True).
Since there's at least one tuple in Y meeting the criteria (S > 1), P2 satisfies the condition. The tuple (P2, Q2, R2) from X will be returned.
-
For P = P3:
In relation Y, the tuple with P=P3 is (P3, 1).
Checking the S value: 1 > 1 (False).
Since no tuple in Y meets the criteria (S > 1) for P=P3, P3 does not satisfy the condition. The tuple (P3, Q3, R2) from X will NOT be returned.
Final Result
The tuples from relation X that satisfy the expression's conditions are those with P values P1 and P2. These are:
- (P1, Q1, R1)
- (P2, Q2, R2)
Therefore, the total number of tuples returned is 2.
