Explanation

The set $A=\{0,1,2,3,4,5,6,7,8,9\}$.

The relation is defined as

$$ (x,y)\in R \iff |x-y|\text{ is a multiple of }3. $$

Partition the set $A$ according to remainders modulo $3$.

$$ \{0,3,6,9\},\quad \{1,4,7\},\quad \{2,5,8\} $$

Elements within the same group differ by a multiple of $3$, hence are related.

Now count ordered pairs in each group.

First group has $4$ elements, so number of ordered pairs:

$$ 4^2=16 $$

Second group has $3$ elements:

$$ 3^2=9 $$

Third group also has $3$ elements:

$$ 3^2=9 $$

Total number of ordered pairs in $R$:

$$ n(R)=16+9+9=34 $$

Hence Statement I, which claims $n(R)=36$, is incorrect.

Now check Statement II.

For all $x\in A$, $|x-x|=0$, which is a multiple of $3$, so $R$ is reflexive.

If $|x-y|$ is a multiple of $3$, then $|y-x|$ is also a multiple of $3$, so $R$ is symmetric.

If $|x-y|$ and $|y-z|$ are multiples of $3$, then $|x-z|$ is also a multiple of $3$, so $R$ is transitive.

Thus $R$ is reflexive, symmetric and transitive.

Hence $R$ is an equivalence relation and Statement II is correct.

Therefore, the correct option is Statement I is incorrect but Statement II is correct.

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