Hello, how to find formula that will express a equivalence relation which has at most three equivalence classes?
For example here is a graph with 3 eq. classes:
I wrote 3 required property's for equivalence relation with this formula:
\(\displaystyle
\displaystyle
\varphi \,\colon = ( \forall x E(x,x)) \land \\
( \forall x \forall y(E(x,y) \to E(y,x))) \land \\
( \forall x \forall y \forall z((E(x,y) \land E(y,z)) \to E(x,z))) \land \\
...
\)
But I dont know how the last part looks like. For example the last part should express that in every 4 random nodes 2 are equivalent, but how to express this with formula or is there some other solution?
Thx.
For example here is a graph with 3 eq. classes:
I wrote 3 required property's for equivalence relation with this formula:
\(\displaystyle
\displaystyle
\varphi \,\colon = ( \forall x E(x,x)) \land \\
( \forall x \forall y(E(x,y) \to E(y,x))) \land \\
( \forall x \forall y \forall z((E(x,y) \land E(y,z)) \to E(x,z))) \land \\
...
\)
But I dont know how the last part looks like. For example the last part should express that in every 4 random nodes 2 are equivalent, but how to express this with formula or is there some other solution?
Thx.