I'm supposed to find all antisymmetric and transitive relations in a 2 element set. I have so far:$$ \text{antisymmetric: } (a,b) ;(b,a); (\emptyset); (a,a),(a,b); (a,a),(b,a); (b,b),(a,b); (b,b),(b,a) $$ $$ \text{transitive: }(a,a),(b,b),(a,b),(b,a);(a,a),(b,b),(a,b);(a,a),(b,b),(b,a); (a,a),(b,b); (a,a);(b,b);(\emptyset) $$
I should have 12 antisymmetric relations and 13 transitive relations. I'm missing some. Can somebody help me out?
Antisymmetric doesn't exactly mean "non-symmetric". For example $\{(a, a), (b, b)\}$ is antisymmetric, and symmetric, as are $\{(a, a)\}$ and $\; \{(b, b)\}$. $\;\{(a, a), (b, b), (a, b)\}$ is also missing from your list, as is $\{(a, a), (b, b), (b, a)\}$. That should give you $12$ antisymmetric relations.
In your second list, what about $\{(a, b)\}$, and $\{(b, a)\}?$. You're also missing $\{(a,a), (a, b)\}$ and $\{(b, b), (b, a)\}$, {(a, a), (b, a)}, and $\{(b, b), (a, b)\}$. That should give you now $13$ transitive relations in all.