Let $$R$$ be a relation on $$N \times N$$ defined by $$a, b R c, d$$ if and only if $$ad(b - c) = bc(a - d)$$. Then $$R$$ is

To determine the nature of the relation $$R$$, we analyze the given condition directly:

$$ad(b - c) = bc(a - d)$$

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Step 1: Check for Reflexivity

For a relation to be reflexive, $$(a, b) R (a, b)$$ must hold for all $$(a, b) \in \mathbf{N} \times \mathbf{N}$$. Substituting $$(c, d) = (a, b)$$ into the given equation:

$$ab(b - a) = ba(a - b)$$

Since multiplication is commutative ($$ba = ab$$), we can rewrite the right side:

$$ab(b - a) = -ab(b - a)$$

$$2ab(b - a) = 0$$

Since $$a$$ and $$b$$ are natural numbers, $$ab \neq 0$$, which leaves:

$$b - a = 0 \implies a = b$$

This statement is only true when $$a = b$$. For any pair where $$a \neq b$$ (such as $$(1, 2)$$), the relation fails.

Therefore, $$R$$ is not reflexive.

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Step 2: Check for Symmetry

For a relation to be symmetric, if $$(a, b) R (c, d)$$ is true, then $$(c, d) R (a, b)$$ must also be true.

We start with the given true equation:

$$ad(b - c) = bc(a - d)$$

Multiplying both sides of the equation by $$-1$$:

$$-ad(b - c) = -bc(a - d)$$

$$ad(c - b) = bc(d - a)$$

Using the commutative property of multiplication ($$ad = da$$ and $$bc = cb$$):

$$da(c - b) = cb(d - a)$$

Rearranging the left and right sides:

$$cb(d - a) = da(c - b)$$

This matches the exact condition required for $$(c, d) R (a, b)$$.

Therefore, $$R$$ is symmetric.

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Step 3: Check for Transitivity

For a relation to be transitive, if $$(a, b) R (c, d)$$ and $$(c, d) R (e, f)$$, then $$(a, b) R (e, f)$$ must be true.

Let us test this using a specific numerical counterexample directly in the given equation format:

Let $$(a, b) = (2, 3)$$, $$(c, d) = (6, 3)$$, and $$(e, f) = (3, 6)$$.

First, test if $$(2, 3) R (6, 3)$$ is true:

$$(2)(3)(3 - 6) = (3)(6)(2 - 3)$$

$$6(-3) = 18(-1) \implies -18 = -18$$ (True)

Second, test if $$(6, 3) R (3, 6)$$ is true:

$$(6)(6)(3 - 3) = (3)(3)(6 - 6)$$

$$36(0) = 9(0) \implies 0 = 0$$ (True)

Now, test if $$(2, 3) R (3, 6)$$ holds:

$$(2)(6)(3 - 3) = (3)(3)(2 - 6)$$

$$12(0) = 9(-4) \implies 0 = -36$$ (False)

Therefore, $$R$$ is not transitive.

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Combining these observations, the relation $$R$$ is symmetric but neither reflexive nor transitive.