Let A = {0 ,1,2,...,9}. Let R be a relation on A defined by (x,y) $$\in$$ R if and only if $$\mid x - y \mid $$ is a multiple of 3.

Given below are two statements:

Statement I: $$n (R) = 36.$$
Statement II: R is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below

Let $$A = \{0, 1, 2, \ldots, 9\}$$. The relation $$R$$ on $$A$$ is defined by $$(x, y) \in R$$ if and only if $$|x - y|$$ is a multiple of 3. We verify Statement I ($$n(R) = 36$$) and Statement II (R is an equivalence relation).

Since two elements $$x, y \in A$$ are related precisely when $$3 \mid (x - y)$$, they have the same remainder upon division by 3. Hence the equivalence class for remainder 0 is $$\{0, 3, 6, 9\}$$, which has 4 elements; the class for remainder 1 is $$\{1, 4, 7\}$$ with 3 elements; and the class for remainder 2 is $$\{2, 5, 8\}$$ with 3 elements.

Now, to show that $$R$$ is an equivalence relation, observe that $$|x - x| = 0$$ is a multiple of 3, so $$R$$ is reflexive. Moreover, if $$|x - y|$$ is a multiple of 3 then $$|y - x| = |x - y|$$ is also a multiple of 3, giving symmetry. Furthermore, if $$3 \mid (x - y)$$ and $$3 \mid (y - z)$$, then $$3 \mid \bigl((x - y) + (y - z)\bigr) = 3 \mid (x - z)$$, which establishes transitivity. Therefore, $$R$$ is an equivalence relation.

Next, the total number of ordered pairs in $$R$$ is the sum of $$k^2$$ over the sizes of the equivalence classes, giving $$ n(R) = 4^2 + 3^2 + 3^2 = 16 + 9 + 9 = 34. $$ Since $$34 \neq 36$$, Statement I is incorrect.

The correct answer is Option (2): Statement I is incorrect but Statement II is correct.