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Using the fact that \begin{align*}(a|b \ \land \ a|c) &\implies a|(b+c) \\ a|b &\implies a| bc \\ (a|b \ \land \ b|c) &\implies a|c \end{align*}

then I want to prove $\equiv_d$ is an equivalence relation on $\mathbb Z$ for every positive integer $d$

I'm pretty confused on this, but so far what I have done is attempt to show reflexivity and symmetry, but not yet transitivity and I don't know how to apply the predicates above. For reflexivity I said $dRd \implies d-d \in \mathbb Z, d\in \mathbb Z^+$, and since $0\in \mathbb Z$ then there is a reflexive relation. For symmetry I said if we take any $k\in \mathbb Z$, then $dRk \implies kRd$. So then $d-k \in \mathbb Z, k-d \in \mathbb Z$. Since $d-k=c, k-d=-c$ then symmetry is satisfied. And it is this point I don't know how to continue further.

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  • $\begingroup$ Are you asking about $\mathbf Z / d\mathbf Z$? $\endgroup$ Nov 1 '20 at 16:42
  • $\begingroup$ I'm not sure, my prof. notes don't specify $\endgroup$
    – Lex_i
    Nov 1 '20 at 16:43
  • $\begingroup$ "Using only the fact that" are you sure you can only use the fact. I don't see any way to prove $a|a$ using only those facts, and we have no way of knowing whether $a|b$ in the first place for any $a|b$. For example, how can you prove $1|1$ or $2|4$ or $5\not \mid 7$ using "only" those facts? $\endgroup$
    – fleablood
    Nov 1 '20 at 16:53
  • $\begingroup$ Well not "only", but as long as I use it then it works $\endgroup$
    – Lex_i
    Nov 1 '20 at 16:54
  • $\begingroup$ $(a-b) + (b-c) = a-c$. $\endgroup$
    – fleablood
    Nov 1 '20 at 17:11
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Reflexive: For every $a \in \mathbb Z$, $a\equiv a\pmod d$.

Pf: $d= 1\cdot d$ so $d|d$. So $d|d\times 0 = 0=a -a$ for any $a\in \mathbb Z$. So $a\equiv a \pmod d$.

Definition of $a\equiv a \pmod d$ is that $d|(a-a)$ or in other words $d|0$. Does $d|0$?

Well $d|d$ so $d|d\cdot 0=0=a-a$. So yes it does. So $d|a-a$ so $a \equiv a \pmod d$ for all $a\in \mathbb Z$. So $\equiv_d$ is reflexive.

Symmetric: For every $a,b \in \mathbb Z$ where $a\equiv b\pmod d$ then $b \equiv a \pmod d$.

Proof: $a\equiv b\pmod p$ means $d|a-b$. So $d|-1(a-b) = b-a$. So that means $b\equiv \pmod d$. SO if $a \equiv b \pmod d$ then $b\equiv a \pmod d$. So $\equiv_d$ is symmetric.

Transitive. If $a \equiv b\pmod d$ and $b\equiv c \pmod d$ then $a\equiv c\pmod d$.

Proof. If $a \equiv b \pmod d$ then $d|(a-b)$. And if $b\equiv c\pmod d$ then $d|(b-c)$.

So $d|(a-b) + (b-c) = a-c$.

So $a\equiv c \pmod d$. So $\equiv_d$ is transitive.

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