Let $X = \{3, 5, 9\}, Y = \{2, 3, 6, 7\}$ and the relation, $R = \{(x, y) | y \leqslant x\}.$
For the relation, I manage to come out with $R = \{(2,3), (2,5), (2,9), (3,3), (3,5), (3,9), (6,9), (7,9)\}.$
However, I have difficulties in determining if $R$ is reflective, symmetric or transitive.
I think $R$ is not reflective because not all loops are present, i.e. only $(3,3)$ is present. However, I saw some online source stated that $\leqslant$ is reflective, which confused me.
I also think that $R$ is not symmetric because not all $(x,y)∈R$ implies $(y,x)∈R$, e.g. $(3,5)$ presents but not $(5,3)$.
As for transitive, I think it is transitive too because $(2,3)$ and $(3,5)$ are in the set, as well as $(2,5)$ too.
I'm not sure if I get all these correct or not. Really appreciate if you all could help me identify these. Thanks!
According to definitions of reflexive relation, symmetric relation and transitive relation it makes no sense to check on these properties.
This because here we are not dealing with a binary relation over a set $X$.
Instead we are dealing with a relation $R$ from a set $A$ to a set $B$ (i.e. $R\subseteq A\times B$) where $A\neq B$.
If you still insist on checking then you could interpret $R$ as a relation over set $A\cup B$.
That trick has no consequences for the check itself if it comes to symmetry and transitivity, but from $A\neq B$ it follows directly that $R$ will not be a reflexive relation on $A\cup B$.