The two relations are equivalent in the relational model sense, even though they appear different at first glance.

In relational database theory, a relation is defined as a set of tuples. Sets are inherently unordered — neither the rows nor the columns have any intrinsic sequence. What matters for equivalence is whether both relations contain exactly the same set of tuples with the same attribute-value pairs.

Consider the tuple content of both tables. Section 1 contains:

• (Roll_no: 9, Sports: Cricket)
• (Roll_no: 13, Sports: Football)
• (Roll_no: 17, Sports: Badminton)
• (Roll_no: 21, Sports: Hockey)
• (Roll_no: 24, Sports: Cricket)

Section 2, when we look past the column order and row arrangement, contains:

• (Sports: Badminton, Roll_no: 17) → (Roll_no: 17, Sports: Badminton)
• (Sports: Cricket, Roll_no: 9) → (Roll_no: 9, Sports: Cricket)
• (Sports: Cricket, Roll_no: 24) → (Roll_no: 24, Sports: Cricket)
• (Sports: Football, Roll_no: 13) → (Roll_no: 13, Sports: Football)
• (Sports: Hockey, Roll_no: 21) → (Roll_no: 21, Sports: Hockey)

Both tables contain exactly the same five tuples. The fact that Section 1 is sorted by Roll_no while Section 2 is sorted by Sports, and that the columns appear in different orders, does not change the underlying data. …

Understanding Relational Equivalence

This question tests a fundamental principle of the relational model: what makes two relations the same?

In relational database theory, a relation (table) is defined as a set of tuples (rows) over a set of attributes (columns). Two critical properties follow from this mathematical definition:

1. Row order is irrelevant

A relation is a set of tuples, and sets have no inherent ordering. The first table is sorted by Roll_no, the second by Sports, but this physical arrangement doesn't change the logical content. Think of it like two decks of cards containing the same cards but shuffled differently — they're still the same deck.

2. Column order is irrelevant

Each tuple is formally a mapping from attribute names to values. Whether you write (Roll_no: 9, Sports: Cricket) or (Sports: Cricket, Roll_no: 9), you're describing the same fact. The column sequence in the physical layout is just a display choice.

Verifying the Equivalence

Let's normalize both tables to a standard form (say, sorted by Roll_no, columns as Roll_no, Sports) and compare:

Section 1 (already in this form):

Section 2 (reordered):

Row-by-row, they match perfectly. Every tuple in Section 1 appears in Section 2, and vice versa. The cardinality (5 rows) is identical, the schema is identical (Roll_no, Sports), and the data content is identical.