$S:=S_1 \cup S_2$ is not convex while $S_1,S_2$ are convex
Solution 1:
Drawing a picture really helps.
$ (3,-3)\in S_1$, $(1,0) \in S_2$ but $\frac 1 2 (1,0)+\frac 1 2(3,-3) \notin S_1\cup S_2$.
$S:=S_1 \cup S_2$ is not convex while $S_1,S_2$ are convex
Solution 1:
Drawing a picture really helps.
$ (3,-3)\in S_1$, $(1,0) \in S_2$ but $\frac 1 2 (1,0)+\frac 1 2(3,-3) \notin S_1\cup S_2$.