Name for relationship where a is related to b iff a and b are in different subsets

Question

Yesterday I was out running in the park, and like many others I always run in a counterclockwise direction around the central lake. There are also strange people who always run in a clockwise direction. I realized that all the runners I can recognize by face run clockwise, because they are the ones I actually meet regularly. I never meet the people who run in the same direction as I do.

So we have a set of runners that can be divided into two distinct subsets, and runner A and runner B can recognize each other iff they are not in the same group.

Is there a name for such a relation?

Answer 1

Let $J$ be the set of joggers, runners, and walkers etc. that run around the lake at the park. Define the relation $S$ on $J$ as

$$jSk\iff\text{ }j\text{ sees }k \text{ while jogging, running, or walking around the lake}.$$

Now a reflexive relation $R$ on a set $A$ is one such that $\forall a\in A,aRa$. Clearly $S$ is not reflexive since a person does not see themselves as they run.

A symmetric relation $R$ is one such that if $jRk$, then $kRj$. Note that $S$ meets this criteria because for each person you see, they see you. Thus $S$ is symmetric.

A transitive relation $R$ is one such that $jRk$ and $kRl$ implies that $jRl$. Note that the relation $S$ is not transitive. $jSk$ implies $kSj$ by the symmetry of $S$. That together with $kSl$ show that $k$ sees both $j$ and $l$. Thus $j$ and $l$ run in the same direction and do not see each other.

Finally, an antisymmetric relation $R$ is one such that $jSk$ and $kSj$ imply that $j=k$. Note that $S$ is not antisymmetric, If $jSk$ and $kSj$, the runners $j$ and $k$ see each other, and thus are not the same person.

Answer 2

As other people have said, it's exactly the opposite of an equivalence relation, so it should be called "not equivalent".

It can be justified as follow:

-The relationship where object have the relationship when they are in the same partition is an equivalent class.

-The relationship where object have the relationship when they are not in the same partition is independence of the object to represent the class: that is if you pick $A,B$ from a class and $C,D$ from a class, the relationship of $A$ to $C$ is the same as that of $B$ to $D$.

-Hence you can factor out the equivalent class and made them into object.

-Once factored out, the relationship where object have the relationship when they are not in the same partition turn into the relationship where object have relationship with everything except itself: ie. "not identical" relationship.

Answer 3

We call such a relation usually 'an' orthogonality relation, e.g.: $$A\perp B:\iff A\cap B=\varnothing$$