Given a finite set S, let the relation
R = {(S1, S2) | |S1| < |S2|, S1, S2 ⊆ S}.
Show whether or not R is reflexive, symmetric, antisymmetric or transitive.
So if S = {1,2}
$R = \{(\emptyset,\{1\}),\,(\emptyset,\{2\}),\,(\emptyset,\{1,2\}),\,(\{1\},\{1,2\}),\,(\{2\},\{1,2\})\}$
and going off of this
Reflexive- No because S1 does not equal S2
Symmetric- No because if you swap $(\emptyset,\{2\})$ for example, the relation doesn't hold true
Transitive- No because $\{(\emptyset,\{1\}),\,(\emptyset,\{2\})$ and doesn't have $\{(\{1\},\{2\})$
Anti-Symmetric-No also because it doesn't satisfy aRb and bRa being true, a = b.
I think I've done this correctly, but I dont have a strong grasp on these relations yet.
This relation is only transitive. Because if $(S_1,S_2) \in R$ and $(S_2,S_3) \in R$ then we have $$|S_1| < |S_2|\ \text {and}\ |S_2| < |S_3|.$$ But it follows that $$|S_1| < |S_3|.$$ So $(S_1,S_3) \in R.$