I have this equivalence relation where $S=\mathbb{R}$ and $a\sim b$
$ \iff a=b$ or $a=-b$
I know this is an equivalence relation and that it is also very simple but I am just confused about how to test for reflectivity, symmetry and transitivity? Like how can I see what $a\sim a$ actually means? Like do I check that subbing $a$ as $b$ it still works or? But surely this would mean $a=-a$ ?
Sorry for the confusion but if anyone has the time to offer some insight I would greatly appreciate it!
$a\sim a$ is a statement. A statement can be either true or false. What that statement is is defined by what $\sim$ means, and in your case, for any arbitrary $a,b$, the statement "$a\sim b$" is the same as the statement "$a=b$ or $a=-b$. Therefore, the statement $a\sim a$ is the statement $a=a$ or $a=-a$.
Notice that there is an or, not an and, connecting the two substatements, one being $a=a$ and the other $a=-a$.
In order for a statement $A$ or $B$ to be true, only one of the statements $A$, $B$ needs to be true. So, following from this, we see that $a\sim a$ is a true statement if one of the statements "$a=a$", "$a=-a$" is true. Clearly, one of them ($a=a$) is true, and therefore, $a\sim a $ is also true. Regardless of whether the other, $a=-a$, is true or not.