Let A = 0,1,2,,9.

Question

Let $A = \{0,1,2,\ldots,9\}$. Let $(x,y)\in R$ iff $|x-y|$ 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:

Accepted Answer

Elements of $A$ split into residue classes mod 3:
- $3n$-type: $\{0,3,6,9\}$ --- 4 elements
- $3n+1$-type: $\{1,4,7\}$ --- 3 elements
- $3n+2$-type: $\{2,5,8\}$ --- 3 elements

$|x-y|$ is a multiple of 3 exactly when $x,y$ are in the same residue class, so
$$n(R) = 4^2+3^2+3^2 = 16+9+9 = 34 \neq 36$$
Statement I is false.

Statement II: Reflexive since $|x-x|=0$ is a multiple of 3. Symmetric since $|x-y|=|y-x|$. Transitive: if $|x-y|=3k$ and $|y-z|=3m$ then $x-z = (x-y)+(y-z)$ is a sum of multiples of 3, hence itself a multiple of 3. So $R$ is an equivalence relation. Statement II is true.

Solutions · 1