Composition of Partially Ordered Sets

Question

Prove or disprove: If $R$ and $S$ are partial orderings on $A$, then $R \circ S$ is a partial ordering on $A$.

I started out defining $R$ and $S$ but I am unsure how to use the definition of partially ordered sets to prove that the composition of the two are also partially ordered. Also one thing I have been thinking is how can you guarantee the composition of these two exist?

Answer 1

Here $R \subseteq AXA$ and $S \subseteq AXA$ are the two relations .. now composition $S \circ R$ is defined by $ (a, b) \in S \circ R \iff \exists$ $x \in A$ such that $(a, x) \in R \land (x, b) \in S$ .. use this as definition .. and after that every thing falls in place automatically ..

Answer 2

Here is a counter example Take $A = \{1, 2, 3 \}$

$R = \{ (1, 1), (2, 2), (3, 3), (1, 2)\}$

$S = \{(1, 1), (2, 2), (3, 3), (2, 3), (3, 1), (2, 1)\}$

$S \circ R = \{(1, 1), (2, 2), (3, 3), (2, 3), (2, 1), (3, 1), (1, 2), (1, 3)\}$

But $S \circ R$ is not antisymmetric So it's not a partial order