# Reflexive, symmetric, transitive, and antisymmetric

Can there be a relation which is reflexive, symmetric, transitive, and antisymmetric at the same time? I tried to find so.

If $$A = \{ a,b,c \}$$. Let $$R$$ be a relation which is reflexive, symmetric, transitive, and antisymmetric.

$$R = \{ (a,a), (b,b), (c,c) \}$$

Is this correct? If I'm wrong, can you help me understand it?

Since if $$(a, b)$$ and $$(b, c)$$ are elements of $$R$$ by transitive there would be $$(a, c)$$, but then there should be $$(b, a)$$, $$(c, b)$$ and $$(c, a)$$ by symmetry, but then it would not be antisymmetric. If I'm not mistaken.

• Rather than simply telling you if you're right or where you're wrong, I would recommend you check methodically so you can be confident in the answer. To test transitivity, if you're concerned about missing something, you can write down all 9 pairs of elements or $R$ and see if they're of the form $(x,y)$ and $(y,z)$ (where some of $x,y,z$ can be the same) and if the corresponding $(x,z)$ is in $R$ too. For symmetry, look at all 3 elements of $R$. For reflectivity, look at all 3 elements of $A$. For antisymmetry, look at all 6 unordered pairs of elements of $R$ to look for $(x,y)$ and $(y,x)$. Sep 25, 2018 at 10:07
• Symmetric and antisymmetric forces the relation to be a subset of the diagonal. Reflexive forces the diagonal to be a subset of the relation. Transitivity doesn't really play a role here, though it follows from the other properties. Sep 25, 2018 at 10:59
• Any reason why none of the answers you received were accepted by you?
– 5xum
Oct 2, 2018 at 6:49
• Sorry for the delay Oct 2, 2018 at 8:49

For any set $$A$$, there exists only one relation which is both reflexive, symmetric and assymetric, and that is the relation $$R=\{(a,a)| a\in A\}$$.
You can easily see that any reflexive relation must include all elements of $$R$$, and that any relation that is symmetric and antisymmetric cannot include any pair $$(a,b)$$ where $$a\neq b$$. So already, $$R$$ is your only candidate for a reflexive, symmetric, transitive and antisymmetric relation.
Since $$R$$ is also transitive, we conclude that $$R$$ is the only reflexive, symmetric, transitive and antisymmetric relation.
Apparently the only solution to your question is the diagonal relation, $$R=\{(x,x)|x\in A \}$$ for any set A.