What are the possible relations on this set?

Question

Let $X=\{x,y\}$ and consider all of the possible relations on $X \times X$

I understand there are 16 possible relations and generally I understand the definitions of reflexive, irreflexive, symmetry, etc... But I'm struggling to write down exactly what the relations here would be.

Please could you give a few examples these relations

Answer 1

For every set $X$, relation over it is arbitrary set $R\subset X \times X$. Then for any $a,b\in X$, notation $aRb$ or $a$~$b$ means that $(a,b)\in R$. Note, that in general expression $a,b\in X$ does not assume that $a\neq b$. Simple calculation ensures, that there are exactly 16 relations: $|X|=2,\ |X\times X|=4,\ |\mathcal{P}(X\times X)|=2^4=16$.

Example relations on $\{x,y\}$

1. Special case: emty set. $\emptyset\subset X\times X$ for any $X$. $\emptyset$ is not reflexive, but it is irreflexive, symmetric and transitive (vacuously). Try to check other properties. More information in this article.

2. Reflexive: note, that every relation $R$ with $(x,x)\in R, (y,y)\in R$ will be reflective. These are: $\{(x,x),(y,y)\},\{(x,x),(y,y),(x,y)\},\{(x,x),(y,y),(y,x)\},\{(x,x),(y,y),(x,y),(y,x)\}$

3. Irreflexive: these are exactly the same as reflexive, with deleted elements $(x,x)$ and $(y,y)$: $\emptyset,\{(x,y)\},\{(y,x)\},\{(x,y),(y,x)\}$.

4. Transitive: there are relations that fail to be transitive, but not much. Consider transitive relation $T$ such that $\{(x,y),(y,x)\}\subset T$. Then by definition: $xTy\land yTx\implies xTx$, and $yTx\land xTy\implies yTy$. This means, that any relation $\{(x,y),(y,x)\}\subset R \subset X\times X$ with $R\neq X\times X$ will fail to be transitive. There are exactly 3 of them. All others (13) are transitive.

Answer 2

The $16$ possible relations are simply all subsets of $\{(x,x),(y,y),(x,y),(y,x)\}$.

Reflexive means that the relation has to include $(x,x)$ and $(y,y)$. There are no other restrictions, so each of $(x,y)$ and $(y,x)$ can either be in or out: $2\times 2$ choices. Irreflexive means that the relation can't include $(x,x)$ or $(y,y)$.

Symmetric means that either $(x,y)$ and $(y,x)$ are both in or both out. In each case there are $4$ options, as again there is no restriction on the other pairs, so there are $8$ in total.

Asymmetric means "not symmetric", i.e. there is some $a\neq b$ such that $(a,b)$ is in the relation and $(b,a)$ is not, or vice versa. Antisymmetric means that this is true whenever $a\neq b$. For relations on $x,y$ this is the same thing - the only possibility is for $a$ and $b$ to be $x$ and $y$ in some order - but for larger sets they are genuinely different. So there are $8$ asymmetric and $8$ antisymmetric (all the ones which are not symmetric).

For transitive your numbers are wrong again, as there would only be $16$ if all relations on this set were transitive. But $\{(x,y),(y,x)\}$ is not a transitive relation ($x\sim y$ and $y\sim x$ but $x\not\sim x$). In fact there are $13$ transitive relations.