Interpret the relations

We are given relations on $\mathbb{N}$:

$R_1 = \{(a,b) \in \mathbb{N}\times \mathbb{N} : |a-b| \le 13\}$

and

$R_2 = \{(a,b) \in \mathbb{N}\times \mathbb{N} : |a-b| \;\text{is a multiple of }13\}$

The printed expression for $R_2$ appears as $|a-b| e 13$, which is clearly intended to mean "$|a-b|$ is divisible by $13$", i.e.

$13 \mid |a-b|.$

Check whether $R_1$ is an equivalence relation

An equivalence relation must be:

• reflexive
• symmetric
• transitive

(i) Reflexive

For every $a \in \mathbb{N}$, $|a-a| = 0 \le 13.$ So $R_1$ is reflexive.

(ii) Symmetric

If $(a,b) \in R_1$, then $|a-b| \le 13.$ But $|b-a| = |a-b|$, so $|b-a| \le 13,$ hence $(b,a) \in R_1$. So $R_1$ is symmetric.

(iii) Transitive

We must check whether $(a,b) \in R_1 \text{ and } (b,c) \in R_1 \implies (a,c) \in R_1.$

Take: $a=1,\quad b=14,\quad c=27.$

Then $|1-14|=13 \le 13,$ so $(1,14) \in R_1$.

Also, $|14-27|=13 \le 13,$ so $(14,27) \in R_1$.

But $|1-27|=26 > 13,$ so $(1,27) \notin R_1$.

Therefore, $R_1$ is not transitive.

Hence, $R_1$ is not an equivalence relation.

Check whether $R_2$ is an equivalence relation

Now interpret $R_2 = \{(a,b): 13 \mid |a-b|\}.$

This is equivalent to saying $a \equiv b \pmod{13}.$

Again check the three properties.

(i) Reflexive

For every $a \in \mathbb{N}$, $|a-a|=0,$ and $13 \mid 0$. So $R_2$ is reflexive.

(ii) Symmetric

If $13 \mid |a-b|$, then since $|b-a|=|a-b|,$ we also have $13 \mid |b-a|$. So $R_2$ is symmetric.

(iii) Transitive

Suppose $(a,b) \in R_2$ and $(b,c) \in R_2$. Then $13 \mid (a-b) \quad \text{and} \quad 13 \mid (b-c).$ Adding, $13 \mid [(a-b)+(b-c)] = a-c.$ Hence $13 \mid |a-c|,$ so $(a,c) \in R_2$.

Therefore, $R_2$ is transitive.

So $R_2$ is an equivalence relation.

Evaluate the options

• A: Both $R_1$ and $R_2$ are equivalence relations → False
• B: Neither $R_1$ nor $R_2$ is an equivalence relation → False
• C: $R_1$ is an equivalence relation but $R_2$ is not → False
• D: $R_2$ is an equivalence relation but $R_1$ is not → True

Comparison with stored answer

Stored correct answer is B, but from the above analysis the correct option is D.

So I disagree with the stored answer.