Correct way to write the set theoretic definition of a relation? I want to write the set theoretic definition of a relation $\preceq$ on a set $X$. So I thought I need to write that we have either $a \preceq b$ or $a \not\preceq b$. However writing $\forall a,b: (a \preceq b) \vee \neg(a \preceq b)$ won't do the job I believe since we have that $P \vee \neg P$ is always true even if we have not defined the value (True of False) of $P$. 
So what is the correct way to write it?
 A: We define a relation, $R$, on a set $X$, as a subset of the Cartesian product $X\times X$.  That is to say, $R\subseteq X\times X$.
We say that $a$ is related to $b$, written as $a~ R~ b$ whenever $(a,b)\in R$.
Note that this definition does not require any additional properties of our relation.  Given additional specific properties, we can define such things as partial orders and equivalence relations.

Common classifications of relations:


*

*A relation $R$ is reflexive if for all $a\in X$ you have $a~R~ a$

*A relation $R$ is symmetric if for all $(a,b)\in R$, you also have $(b,a)\in R$

*A relation $R$ is anti-symmetric if for all $(a,b)\in R$ with $a\neq b$, you have $(b,a)\not\in R$.  (equivalently worded as if $a~R~b$ and $b~R~a$ then $a=b$)

*A relation $R$ is transitive if for all $(a,b)\in R$ and $(b,c)\in R$, you also have $(a,c)\in R$.

Useful types of relations:


*

*An equivalence relation is one which is reflexive, symmetric, and transitive.  The prototypical example is $=$.

*A partial order is a relation which is reflexive, anti-symmetric and transitive.  Note, that with a partial order, we do not require that every pair be comparable.

*A total order is a relation which is reflexive, anti-symmetric, transitive, and has the property that you always have either $a~R~b$ or you have $b~R~a$.  The prototypical example is $\leq$.

