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.

  • 2
    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 $(y,x)$. – Mark S. Sep 25 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. – John Coleman Sep 25 at 10:59
  • Any reason why none of the answers you received were accepted by you? – 5xum Oct 2 at 6:49
  • Sorry for the delay – Shehan Tearz Oct 2 at 8:49
up vote 12 down vote accepted

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.

Your answer is correct and you can easily generalize it to a set with more elements

Apparently the only solution to your question is the diagonal relation, $$R=\{(x,x)|x\in A \}$$ for any set A.

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