Check if a relation is reflexive, symmetric and transitive I want to better understand how this actually works, as my solutions are
sometimes not 100% correct.
I have the following relation:

Check if the following relation is reflexive, symmetric, and/or
transitive:
$$ R_1 = \{ (x,y) \mid x,y \in \mathbb{R}, x=0 \land y \geq 0 \}. $$

so by that
$$ R_1 = \{ 00, 01, 02,03,04,05,06,07,08,09,010, \dots, 0R_+ \} $$
Basically $R_1$ is 0 and any $R_+$ number.
It is not reflexive, as $(a,a)$ is not in $R_1$.  I only have 00, but
not 11 or 22 and so on.
It is also not symmetric, as I don't have 01 and 10 or 02 or 50 and
05. So $xRy$ and $yRx$ are not true for $R_1$.
As for the transitivity, well, if $xRy$ and $yRz$ then $xRz$.  Well,
this one is hard to understand. I could use 00 as an example: If $y = 0$
and $z \geq 0$ then $xRz$ would work.  So I would say it is transitive.
Can anybody confirm if this would be correct? If not, i would really
appreciate a correct approach then for this task.
&quot;If xRy and yRz&quot;: Look at the definition of R. You have x = 0 and y ≥ 0 because xRy. You have y = 0 and z ≥ 0 because of yRz. If you have xRy and yRz then you have x = 0 and z ≥ 0, therefore xRz. So R is transitive.
In total you know that x = 0, y = 0 and z ≥ 0, so you also have yRx, although nobody cares much about that.