Defining a relation that is antisymmetric, but not symmetric? Say I have a set = {1,2,3}.
I am trying to think about how I could define a set on X which is antisymmetric but not symmetric. 
At first I had thought the set would be Z = {(1,1),(2,2),(3,3)} but am I correct in thinking that this is symmetric? 
Would the set Z = {(1,1),(2,2),(3,3),(1,2),(2,3)} be both antisymmetric, but not symmetric? Since the (1,1),(2,2) and (3,3) make it antisymmetric, but the fact that (2,1) and (3,2) are missing making it not symmetric?
Thanks
 A: Suppose $R$ is a relation on a set $E$ which is both symmetric and antisymmetric.
Take $a\in E$. Assume you can find $b\in E$ such that $aRb$. By symmetry you get $bRa$. Hence by antisymmetry $a=b$. The same thing holds with $bRa$.
Whence an element is, at most, in relation with itself. So the diagonal set and its subsets are the only example of relation being both symmetric and antisymmetric.
A: Let $A = \{1,2,3\}$. We want a relation $R$ on $A$ that is antisymmetric, but not symmetric. This means that


*

*antisymmetry: for every $(a,b) \in R$ where $a \neq b$, we must also have $(b,a) \notin R$, and

*not symmetric: there must exist some $(a,b) \in R$ such that $(b,a) \notin R$.


Your suggestion is $R = \{(1,1),(2,2),(3,3),(1,2),(2,3)\}$. Let us check if this satisfies the criteria. There are two pairs that satisfies $(a,b) \in R$ with $a \neq b$, namely $(1,2)$ and $(2,3)$. We see that $(2,1) \notin R$ and $(3,2) \notin R$, so it is antisymmetric. Can we say that it is not symmetric? Yes, because $(1,2) \in R$ but $(2,1) \notin R$.
We can make a smaller example by taking some small antisymmetric relation and adding an element to make it not symmetric. We could take $R = \{(1,1)\}$ as an antisymmetric set, and then introduce for example $(1,2)$ to make it not symmetric, so that $R = \{(1,1),(1,2)\}$ is still antisymmetric because $(2,1) \notin R$ and it is not symmetric because $(1,2) \in R$, but $(2,1) \notin R$.
A: How about $$Z=\{(1,2), (2,3),(3,1)\}$$
A: How about '$\le$' (the less than or equal relation)?
