Let $G$ and $H$ be non empty subsets of $\mathbb{R}$, where $G$ is connected and $G\cup H$ is not connected. Then which is true?

Question

Let G and H be non empty subsets of R, where G is connected and G∪H is not connected. Then which is true?

A. If $G\cap H=\varnothing$, then $H$ is connected.
B. If $G\cap H=\varnothing$, then $H$ is not connected.
C. If $G\cap H\neq\varnothing$, then $H$ is connected.
D. If $G\cap H\neq\varnothing$, then $H$ is not connected.

My attempt:
If $G$ is connected and subset of $\mathbb{R}$ then surely $G$ is an interval $[a,b]$.
If I take $H$ to be connected then $G\cup H$ will also be connected if $G\cap H\neq\varnothing$, but the question says that $G\cup H$ is not connected, therefore can I say that $H$ is to be taken-not connected??

Answer 1

Like you said, if $H$ is connected and $G\cap H\neq \varnothing$ then $G\cup H$ is connected. Therefore, if $G\cup H$ is not connected, and $G\cap H\neq\varnothing$, then $H$ must not be connected. (In general, if "$P$ and $Q$" implies $R$, then $\neg R$ implies "$\neg P$ or $\neg Q$". Therefore in this case $\neg R$ and $Q$ implies $\neg P$.)

Therefore only option $D$ is correct. It's not hard to come up with counterexamples for the rest.

EDIT Here are some counter examples:

A: $G=(0,1)$, $H=\{0,1\}$. Then $G$ is connected, $G\cup H=[0,1]$ is connected and $G\cap H =\varnothing$, but $H$ is not connected.

B: $G=[0,1)$, $H=\{1\}$. Then $G$ is connected, $G\cup H=[0,1]$ is connected and $G\cap H =\varnothing$, but $H$ is connected.

C: $G=[0,1]$, $H=\{0,1\}$. Then $G$ is connected, $G\cap H\neq \varnothing$ and $G\cup H=[0,1]$ is connected, but $H$ is not connected.