GATE 2026 DA Q.61: solution and answer

GATE 2026 · DAQ.61 · 2 marks · NAT

Consider an ER model with the entities E1(A11, A12, A13) and E2(A21, A22, A23), where A11, A12, A13 are the attributes of E1, and A21, A22, A23 are the attributes of E2. Let A22 be a multi-valued attribute. A11 and A21 are the primary keys of E1 and E2, respectively.

Let R12 be a many-to-many relationship between E1 and E2. Participation of both E1 and E2 in R12 is total.

The minimum number of relations required to convert the ER model into relational model (assuming there is no other functional dependency) where each relation is in third normal form (3NF) is ______. (Answer in integer)

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Answer and solution

ANSWER

The ER model needs a minimum of 4 relations.

Official answer: 4Source: Official answer key, GATE 2026 DA (IIT Guwahati)

Entity $E 1$ has the key $A_{11}$ and only single-valued attributes, so it becomes one relation $E 1 (A_{11}, A_{12}, A_{13})$ .
Entity $E 2$ has the key $A_{21}$ and one multi-valued attribute $A_{22}$ . A multi-valued attribute cannot be stored as a column of $E 2$ , because a cell would hold a set and break first normal form. So $E 2$ becomes $E 2 (A_{21}, A_{23})$ .
The multi-valued attribute $A_{22}$ gets its own relation $E 2_A 22 (A_{21}, A_{22})$ . Its key is the pair $(A_{21}, A_{22})$ , and $A_{21}$ is a foreign key to $E 2$ .
The many-to-many relationship $R 12$ needs its own relation $R 12 (A_{11}, A_{21})$ . Its key is the pair $(A_{11}, A_{21})$ , and both parts are foreign keys. A many-to-many relationship cannot be folded into $E 1$ or $E 2$ , because that would again store a set of values in one cell.
Total participation of $E 1$ and $E 2$ in $R 12$ only makes a foreign key non-null when the relationship is merged into an entity relation, as in the 1:1 or 1:N case. It does not remove the separate relation for a many-to-many relationship, so it changes nothing here.
The question says there is no other functional dependency. So in each relation the only determinant is its key, and every relation is in 3NF (indeed BCNF).
Count: $E 1$ , $E 2$ , $E 2_A 22$ and $R 12$ give $4$ relations. None can be dropped without losing the multi-valued attribute, the relationship, or 3NF.

KEY IDEA

In ER-to-relational mapping, each entity gets a relation, each multi-valued attribute gets its own relation, and each many-to-many relationship gets its own relation. Total participation only affects foreign-key nullability when a relationship is merged, so it does not change the count here.

WATCH OUT FOR

Counting only 3 relations by folding the multi-valued attribute $A_{22}$ into $E 2$ , or by merging $R 12$ into an entity relation because participation is total. Both break first normal form. Another slip is to count only the two entities plus the relationship and miss the separate relation for $A_{22}$ .

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 1 ER model questions, with notes