well, so far I know there exist no injective map from $S^n\rightarrow R^n$(due to Borsuk-Ulam), so in the case of $3.8$ they are asking are there different point on $S^1$ whic maps to same point in $\mathbb{R}$? so by Borsuk Ulam theorem I can say "Yes", If the $f$ is constant then $A$ is uncounatble, but I have no idea about if the $f$ is non-constant.
for $3.9$ I can say same argument right?
I will be happy about responses. Thank you.