two different relations on a common set question (discrete)

Question

If you have two separate relations, $S$ and $R$, on a common set $A$ and $S \subseteq R$

a.) is R reflexive if S is.

b.) is S antisymmetric if R is.

c.) is R transitive if S is.

Wouldn't these all be true because $S$ is a subset of $R$?

Answer 1

$(a)$ and $(b)$ would be true.

For $(c)$, let's consider $A=\{x,y,z\}$, $S=\{(x,x),(y,y),(z,z)\}$ and $R=S\cup \{(x,y),(y,z)\}$. $S$ is clearly transitive, but $R$ isn't.