Let 𝑅 be a binary relation on the set {1, 2, … , 10}, where (𝑥, 𝑦) ∈𝑅 if the product
of 𝑥 and 𝑦 is square of an integer. Which of the following properties is/are satisfied
by 𝑅?

Question

A database has 25,000 fixed-length records of 100 bytes (primary key 15 bytes). The data file is block-aligned so each record is fully contained within a block. The file is indexed by a primary index file, also block-aligned and ordered. Block size is 1024 bytes and a block pointer is 5 bytes. Each index entry stores the block’s anchor key (15 bytes) and a pointer (5 bytes). Binary search on an index file of \(b\) blocks needs \(\lceil \log_2 b \rceil\) block accesses in the worst case. Given a key, the number of block accesses required to identify the data file block that may contain the record, in the worst case, is ____________.

Answer 1

A quicker way to analyze this relation is to treat it as a partition problem instead of checking each property directly from the definition.

Step 1: Group the numbers 1 to 10 by their square-free part (strip out the largest square factor from each): 1 = 1 (square-free 1), 2 = 2, 3 = 3, 4 = \(2^2\) (square-free 1), 5 = 5, 6 = 6, 7 = 7, 8 = \(2 \times 2^2\) (square-free 2), 9 = \(3^2\) (square-free 1), 10 = 10.

Step 2: So the groups by square-free value are: {1,4,9} share square-free 1, {2,8} share square-free 2, while 3, 5, 6, 7, 10 are singleton groups.

Step 3: \((x,y) \in R\) exactly when \(x\) and \(y\) fall in the same group - this is precisely how an equivalence relation induced by a partition behaves.

Step 4: Every equivalence relation is automatically reflexive, symmetric and transitive by construction, since it partitions the set into disjoint classes and only relates elements within the same class.

Step 5: Antisymmetry fails whenever a group has more than one element, since two distinct elements of the same group relate to each other in both directions. Groups {1,4,9} and {2,8} both have more than one element, so antisymmetry breaks down.

Conclusion: The relation is reflexive, symmetric and transitive, but not antisymmetric - final answer A, B, C.

Let 𝑅 be a binary relation on the set {1, 2, … , 10}, where (𝑥, 𝑦) ∈𝑅 if the product
of 𝑥 and 𝑦 is square of an integer. Which of the following properties is/are satisfied
by 𝑅?

Show Hint

The Correct Option is A, B, C

Solution and Explanation

Top Questions on Relations: Reflexive, Symmetric, Transitive, Antisymmetric

Questions Asked in GATE CS exam

Let 𝑅 be a binary relation on the set {1, 2, … , 10}, where (𝑥, 𝑦) ∈𝑅 if the productof 𝑥 and 𝑦 is square of an integer. Which of the following properties is/are satisfiedby 𝑅?

Show Hint

The Correct Option is A, B, C

Solution and Explanation

Top Questions on Relations: Reflexive, Symmetric, Transitive, Antisymmetric

Questions Asked in GATE CS exam

Let 𝑅 be a binary relation on the set {1, 2, … , 10}, where (𝑥, 𝑦) ∈𝑅 if the product
of 𝑥 and 𝑦 is square of an integer. Which of the following properties is/are satisfied
by 𝑅?