Let R be a binary relation on the set {1,2,…,10}, where (x,y)∈R if the product…
2026
Let R be a binary relation on the set {1,2,…,10}, where (x,y)∈R if the product of x andy is a perfect square.
Which of the following properties are satisfied by R?
- A.
Reflexive
- B.
Symmetric
- C.
Transitive
- D.
Antisymmetric
Attempted by 86 students.
Show answer & explanation
Correct answer: A, B, C
Given a binary relation R on the set {1, 2, ..., 10} where (x, y) ∈ R if x * y is a perfect square.
Reflexive: For any x, x * x = x^2, which is always a perfect square. Thus, (x, x) ∈ R for all x. The relation is Reflexive.
Symmetric: If x * y is a square, then y * x is the same product and also a square. Thus, (x, y) ∈ R implies (y, x) ∈ R. The relation is Symmetric.
Transitive: If x * y = a^2 and y * z = b^2, then (x * y) * (y * z) = (ab)^2. Dividing by y^2 gives x * z = (ab/y)^2. Since x, y, z are integers, x * z is a square. The relation is Transitive.
Antisymmetric: Consider x = 2 and y = 8. x * y = 16 (square), so (2, 8) ∈ R. Also (8, 2) ∈ R. Since 2 ≠ 8, the relation is not Antisymmetric.
Conclusion: The relation R satisfies Reflexive, Symmetric, and Transitive properties.