For$ a, b \in \mathbb{Z}$ define $a\sim b$, if and only if $a^2 – b^2$ is divisible by $3$.

Question

$a)$ Prove that $\sim$ is equivalent relation on $\mathbb{Z}$.

$b)$ What is the $0$? What is the $1$?

$c)$ What is the partition of $\mathbb{Z}$ determined by this equivalence relation?

For a I'm supposed to use reflexive, symmetric and transitive properties but my professor used over simplified examples and I'm not sure how to apply it to this problem.

I don't understand what $b)$ is asking for and how to solve it.

For $c)$, I can only understand partitions visually, so how do i put it into words?

Thanks in advance

Answer 1

(a)

Check to make sure that the relation is indeed reflexive, symmetric, and transitive.

Reflexivity: Suppose that $a\in \Bbb Z$. Does that imply that $a\sim a$? That is to say, is it true that for every $a\in \Bbb Z$ that $a^2-a^2$ is divisible by $3$?

Yes, of course, since $a^2 - a^2 = 0 = 0\cdot 3$ is an integer multiple of three, that implies that $a\sim a$ in this relation.

Symmetry: Suppose that $a,b\in\Bbb Z$ such that $a\sim b$. Is it true then that $b\sim a$? That is to say, supposing that $a^2-b^2$ is divisible by $3$, does it follow that $b^2-a^2$ is also divisible by $3$?

Yes, of course, since if $a^2-b^2=3\cdot k$ for some integer $k$ it follows that $b^2-a^2 = 3\cdot (-k)$ and so $b^2-a^2$ is also an integer multiple of $3$ just like $a^2-b^2$ was, implying that indeed $b\sim a$

Transitivity: Suppose that $a,b,c\in\Bbb Z$ (not necessarily distinct!) such that $a\sim b$ and $b\sim c$. Does it follow that $a\sim c$? That is to say, supposing that $a^2-b^2$ is a multiple of three and $b^2-c^2$ is also a multiple of three (possibly different than the first) does it follow that $a^2-c^2$ is also a multiple of three?

Let as exercise to reader

(b)

I assume that what they mean to ask is "What is the equivalence class of $0$?"

The equivalence class of $0$ is the set of all elements in the domain such that they are related to zero. Commonly written as:

$$[0] = \{a\in\Bbb Z~:~a\sim 0\}$$

Try to see if you can spot a pattern. Is $0\sim 1$? Is $0\sim 2$? Is $0\sim 3$?

Similarly, $[1]=\{a\in\Bbb Z~:~a\sim 1\}$. Is $1\sim 2$? Is $1\sim 4$?