R is a partial order on A, and $B\subseteq A$, prove that R is also a partial order on B
I've been doing exercises on relations and it bothers me a lot if this holds, hence my question.
Reflexivity: Let $x$ be an arbitrary element of B. But if so $x$ is also in A; and since R is reflexive on A, i.e. $\forall x\in A$ $xRx$, R is also reflexive on B.
Anti-symmetric: Let $x$ and $y$ be arbitrary elements of B, and assume $xRy$ and $yRz$. Since R is anti-symmetric on A, and $x$ and $y$ are in A in virtue of the subset property, we can utilise our assumption and this fact to deduce $x=y$.
Transitivity: Let $x$, $y$ and $z$ be arbitrary elements of B and assume $xRy$ and $yRz$. Again by the same approach, in virtue of the subset property and R being transitive on A, we can apply our assumptions to get $xRz$.
I know it's a ridiculously simple question, but could anyone just confirm if my workings are right please? Thank you so much!