I know that an equivalence relation (~), by definition, has to hold 3 properties.
1.) Reflexive
2.) Symmetric
3.) Transitive
Although actually using this proof technique is very confusing to me. I am looking for a detailed explanation and a possible resource such as a website or video that has discrete mathematics explanations. Looking at the 3 properties to an equivalence relation I understand what all of them mean but I don't quite understand how to show the same meaning within another proof.
All help is much appreciated!
Best Answer
I'm not sure if this answers everything, but hopefully this helps.
Equivalence relations are, in a sense, a generalization of equality. So we ask ourselves: what are the defining properties of equality. Well first, we notice that there always holds $x=x$ (reflexivity). Secondly, we have that if $x=y$ then $y=x$ (symmetry). Finally, if $x=y$ and $y=z$ then $x=z$ (transitivity).
Now, I will address your problem more specifically, I am assuming that you are defining "$\sim$" to be such that $x\sim y$ iff there exists a number $n\in\mathbb{Z}$ such that $x = y + 7n$.
Using this definition, it is clear that $x\sim x$. Indeed, picking $n=0$ we see that $x = x + 7\cdot 0$. This shows reflexivity.
For symmetry, suppose that $x \sim y$. Then there exists $n\in \mathbb{Z}$ such that $x = y+7n$. But then, we also have $y = x + 7\cdot (-n)$. Ergo, $y\sim x$.
Finally, suppose that $x\sim y$ and $y\sim z$. Then there exist $n_1, n_2\in\mathbb{Z}$ such that $x = y+7n_1$ and $y = z+7n_2$. It follows that $$ x = y+7n_1 = z+7n_2+7n_1 = z+7(n_1+n_2) $$ which means that $x\sim z$.
I hope this answers at least part of your question!