Is $a\sim b$ exactly when $a \times b$ is divisible by $3$ an equivalence relation?

Question

Let $\sim$ be define so that $a\sim b$ exactly when $a \times b$ is divisible by $3$. Is this an equivalence relation? If not, which of the three properties (reflexive, symmetric, transitive) does not hold?

Solution:

We need to test each of the following cases to see if they hold.

Here are my assumptions: $a\times b$ is divisible by $3$ in the following cases:

• Case 1: $3\mid a $, and $3\nmid b$. Example: $a=3, b=2, a\times b=6$
• Case 2: $3\nmid {a}$, and $3\mid b$. Example: $a=2, b=3, a\times b=6$
• Case 3: $3\mid a$, and $3\mid b$. Example: $a=3, b=3, a\times b=9$

Reflexive Test:

• $aRa: = \{(a,a): 3\mid a\times a\}$
• $bRb: = \{(b,b): 3\mid b\times b\}$

This fails in $aRa$ when $a$ is not divisible by $3$ according to case 1. This also fails in $bRb$ when $b$ is not divisible by $3$ according to case 2.

Symmetric Test:

• $aRb: = \{(a,b): 3\mid a\times b\}$
• $bRa: = \{(b,a): 3\mid b\times a\}$

This works because if $a\times b$ is divisible by $3$, then $a$ is divisible by $a\times b$ and $b$ is divisible by $b\times a$.

Transitive Test: Honestly I am not exactly sure how to describe the relation here, since we need $a$, $b$ and $c$.

What is $c$ in this case, is it the result of $a\times b$?

Am I on the right track?

Answer 1

I think you're making things more complicated by using notation, rather than thinking about what the question means.

Reflexivity. Is it the case that, for every number $a$, $a^2$ is divisible by $3$? If yes, then the relation is reflexive. If no, then the relation is not reflexive.

Symmetry. Is it the case that, if $ab$ is divisible by $3$, then $ba$ is divisible by $3$? If yes, then the relation is symmetric. If no, then the relation is not symmetric.

Transitivity. Is it the case that, if $ab$ is divisible by $3$ and $bc$ is divisible by $3$, then $ac$ is divisible by $3$? If yes, then the relation is transitive. If no, then the relation is not transitive.

Do you see the answers to these questions?

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Note that these questions are just translations, into words, of the usual formal definition. E.g., $R$ is reflexive iff for every $a$, we have $aRa$. In this case, "$xRy$" means "$xy$ is divisible by $3$," so the question "Is $R$ reflexive?" is the same as the question "Is it the case that, for every $a$, $a^2$ is divisible by $3$?"

Answer 2

$3$ is prime, so $a\sim b \iff 3|ab \iff 3|a \text{ or } 3|b$.

$\sim$ is not transitive: $1\sim 3$ and $3\sim 1$, but not $1\sim 1$.

As just seen, $\sim$ is not reflexive.

It is symmetric, obviously.

Answer 3

It is not transitive! Take $3|(2\times 3),~~3|(3\times 4)$ but $3$ does not divide $2\times 4$.