For my question, please see the attached image (specifically, example 5).
For example 8 (not shown), we're supposed to refer back to example 5 (shown below) and see which of the given relations are reflexive. We can easily see that $R_1$ and $R_4$ are reflexive, as for every a $\in$ R, (a, a) $\in$ R.
I'm more confused by the fact that $R_3$ is also reflexive. Part of the condition makes sense - specifically, $R_3$ = {(a, b) | a = b}. But we also know that $R_3$ = {(a, b) | a = -b}. This means that in addition to containing the pairs (1, 1) and (2, 2), it should contain (1, -1). But there isn't an a $\in$ R such that a = -1.
Is it reflexive because $a = -b \Rightarrow (1) = -(-1) \Rightarrow 1 = 1$, and there is the pair (1, 1) $\in$ R?
Thank you in advance!
$(1,1)$ is in $R$, and so is $(1,-1)$, correct.
For reflexivity, however, we don't care whether or not $(1,-1)$ is in $R$. So, the fact that we have some $(a,b)$ with $a \neq b$ in $R$ does not make $R$ non-reflexive.
Now, we do care whether $(-1,-1)$ is in $R$, but it is: for $a = -1$, you can pick $b=-1$.
And indeed, for any integer $a$, we can pick $b=a$, and so you have $(a,a)$ in $R$. This is why $R$ is reflexive.