If $R$ and $S$ are endorelations then are the following statement true or false
(1) If $R$ and $S$ are transitive, then so is $R \circ S$
(2) If $R$ and $S$ are irrefexive, then so is $R \circ S$
(3) If $R$ and $S$ are equivalence relations, then so is $R \circ S$
I was able to conclude that (2) is false by giving a counter example. i.e. If $R=\{(0,1)\}$ and $S=\{(1,0)\}$ then $R \circ S=\{(0,0)\}$ which is reflexive. Hence (2) is false
Does anybody have same counter examples or proofs for the (1) and (3)
You can kill (1) and (3) with one blow by giving two equivalence relations whose composition is not transitive. Consider these equivalence relations $R$ and $S$ on the set $\{0,1,2\}$: $$R=\{(0,0),(1,1),(2,2),(0,1),(1,0)\}$$ $$S=\{(0,0),(1,1),(2,2),(1,2),(2,1)\}$$ In fact $R\circ S$ is neither transitive nor symmetric. This follows from the facts:
$(2,1)\in R\circ S\ $ [since $(2,2)\in R$ and $(2,1)\in S$];
$(1,0)\in R\circ S\ $ [since $(1,0)\in R$ and $(0,0)\in S$];
$(0,2)\in R\circ S\ $ [since $(0,1)\in R$ and $(1,2)\in S$];
$(2,0)\notin R\circ S.$