Problem
I have these two relations over $A$, and I am supposed to determine whether they are reflexive, symmetric, antisymmetric, and/or transitive. I have determined that they are not reflexive or symmetric, however I'm unsure whether they're both antisymmetric, and I don't understand how to determine whether or not they're transitive. Will someone please explain?
Let $A = \{1,2,3,4,5,6\}$
$R_1 = \{(x,y) | \lceil log_2x]\rceil < \lceil log_2y\rceil \}$
$R_2 = \{(x,y) | \lceil log_2x]\rceil = 2 + \lceil log_2y\rceil \}$
Attempt
I see that for BOTH $(x,y) \rightarrow \neg(y,x)$ so my guess is that they're both antisymmetric. Is that correct? If so, which one is transitive? Is it possible for a relation to be both symmetric and transitive?
Ordered pairs which satisfy each relation
$R_1: (1,2) (1,3) (1,4) (1,5) (1,6) (2,3) (2,4) (2,5) (2,6) (3,5) (3,6) (4,5) (4,6)$
$R_2: (3,1) (4,1) (5,2) (6,2)$