What binary relation is neither symmetric, nor asymmetric nor antisymmetric? I thought it was the relation $\varnothing$, but the answer in the textbook I am using does not mention this as a possible answer. I don't understand why it can't be the answer. Could anyone explain?
 A: $\varnothing$ is the (unique) example of a relation that is symmetric, asymmetric and antisymmetric (so the opposite of what you are looking for).
E.g. a relation $R$ is not symmetric if an ordered pair $\langle a,b\rangle$ exist with $\langle a,b\rangle\in R$ and $\langle b,a\rangle\notin R$. 
This is evidently not true for $R=\varnothing$ (it contains no ordered pairs) so we conclude that $\varnothing$ is symmetric.
Likewise it can be shown that $\varnothing$ is asymmetric and antisymmetric. 
The implications involved are vacuously true.
A: In the empty relation, it is impossible for $xRy$, so that whenever $xRy$ it is true that $yRx$ - you have a statement about elements of the empty set, and there aren't any such elements to make the statement false.
A: For a relation $R$ to be symmetric, every ordered pair $(a,b)$ in $R$ will also have $(b,a)\in R$.
For a relation to be asymmetric, every ordered pair $(a,b)\in R$ does not have $(b,a)\in R$.
For a relation to be antisymmetric, if both $(a,b)$ and $(b,a)$ are in $R$ then $a=b$.
So we want $R$ such that for some $a\neq b$, $(a,b)$ and $(b,a)$ are both in $R$ this makes sure $R$ is neither asymmetric nor antisymmetric; but at the same time we want some $(c,d)\in R$ such that $(d,c)\notin R$, as this will ensure that $R$ is not symmetric.
In either case, we need witnesses in $R$ to prove that it is not symmetric or asymmetric. Therefore it cannot be empty.

I will leave the grueling details of writing down such $R$ for you.
A: Example:
S = {a,b,c}
R = { (a,b), (a,c), (c,a) }
Then:
1. R is not symmetric, because aRb is true, but bRa is false.
2. R is not asymmetric, because aRc and cRa.
3. R is not antisymmetric, because aRc and cRa, yet a<>c.
