In order to determine which of the relations listed is reflexive, we must recall that a relation R on a set A is reflexive if for every element x ∈ A, the pair (x, x) is in R. This means that x must be related to itself.

Let’s examine each of the relations listed:

(a) R = {(x, y) : x > y, x, y ∈ ℕ}

For this relation to be reflexive, we must have x > x for all x ∈ ℕ. However, this is never the case since x is never greater than itself. So this relation is not reflexive.

(b) R = {(x, y) : x + y = 10, x, y ∈ ℕ}

For this relation to be reflexive, we require that x + x = 10 for all x ∈ ℕ. This yields 2x = 10, or x = 5. So, the only element that satisfies this is x = 5. Hence, the relation is **not reflexive** for all elements of ℕ, but only for x = 5.

(c) R = {(x, y) : xy is a square number, x, y ∈ ℕ}

For this relation to be reflexive, we must have x * x to be a square number for all x ∈ ℕ. Since x * x = x² is always a square number for all natural numbers x, this relation is reflexive.

(d) R = {(x, y) : x + 4y = 10, x, y ∈ ℕ}

For this relation to be reflexive, we require x + 4x = 10 for all x ∈ ℕ. This simplifies to 5x = 10, or x = 2. Hence, the only element satisfying this condition is x = 2, so this relation is not reflexive for all elements of ℕ, only for x = 2.

Conclusion:
The only reflexive relation is:
– (c) R = {(x, y) : xy is a square number, x, y ∈ ℕ}.

Click here for more:
https://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-1