Why are all singletons confluent? A relation R of a set M is confluent, if $ \forall x \in M \forall w1,w2 \in M :((xRw1 \land xRw2 ) \to \exists z \in M (w1Rz \land w2Rz)) $ . 1\. Someone told me that all singletons, no matter the relation, are confluent. Why? Same reason as why empty set is confluent (cant test the statement)? 2\. Identity relations are always confluent, why? Because x=y=z always works?

1. A relation _on_ a singleton set is always confluent. If $M=\\{a\\}$ all we need to check is that $$(a\mathrel R a \land a\mathrel R a) \to \exists z \in M : (a\mathrel R z\land a \mathrel R z)$$ which is obviously true -- if the left-hand side is true, then $a$ itself works as $z$.

2. If $R$ is the identity relation, the left-hand side is $x=w_1\land x=w_2$, so the only relevant case to check the right-hand side for is $x=w_1=w_2$. In that case choosing $z=x$ obviously makes $w_1=z\land w_2=z$ true.