Reflexive and transitive closure of a binary relation

Question

If relation A is a binary relation between terms of the form (C,s), and relation B is the reflexive and transitive closure of A, could somebody briefly explain what it means to be a 'Reflexive and transitive closure'?

Thanks.

Answer 1

Suppose $A$ is a binary relation on $S$. Then the transitive reflexive closure $A^*$ of $A$ is the smallest relation which includes $A$ and which is transitive and reflexive. In order to get $A^*$ from $A$, we simply add all the elements that are needed in order to make it transitive and reflexive. Formally, this means that $A^*$ is given by $$A^* = A \cup \{(x,x) \mid x \in S\} \cup \{(x,y) \mid (x,x_1),(x_1,x_2),\dots,(x_n,y) \in S \text{ for some } n \geq 1\}.$$