Correct way to write the set theoretic definition of a relation?

Question

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?

Answer 1

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.

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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 $(a,c)\in R$.

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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 $\leq$.