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The question is based on the information provided below.

A number of dacoits need to hole up in their hideout. The problem is that, as with many criminal groups, there are a number of personality conflicts that disallow certain members of the dacoit group from being in the hideout at the same time as certain other members. The personality issues of the bandits $\text{A, B, C, D, E, F, G,}$ and $\text{H}$ are illuminated by the following constraints:

1. If $\text{D}$ is in the hideout, then $\text{E}$ is in the hideout.
2. If $\text{E}$ is not in the hideout, then $\text{A}$ is in the hideout.
3. If $\text{B}$ is not in the hideout, then $\text{C}$ is not in the hideout.
4. If $\text{F}$ is in the hideout, then $\text{H}$ is not in the hideout.
5. If $\text{A}$ is not in the hideout, then $\text{B}$ is in the hideout.

If $\text{D}$ and $\text{C}$ are in the hideout, then who could be outside of the hideout?

1. $\text{A}$ and $\text{E}$
2. $\text{B}$ and $\text{G}$
3. $\text{E}$ and $\text{B}$
4. $\text{H}$ and $\text{F}$

Explanation:

If $D$ and $C$ are in the hideout, then who could be outside of the hideout?

(D) According to diagram (A.) $E$ and $B$ must be in the hideout. Therefore, $A$, $G$, $F$, or $H$ could be out.

(A) $E$ must be in.

(B) $B$ must be in.

(C) $E$ and $B$ must be in.

(E) D is the correct answer.

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878 points
We have

$(D\to E), (\neg E \to A), (\neg B \to \neg C), (F \to \neg H), (\neg A \to B)$

Now, we have $(D,C)$ inside the hideout.

So, $E$ must also be inside (due to $D \to E),$ and $B$ must also be inside due to $([\neg B \to \neg C] \leftrightarrow [C \to B]).$ This eliminates options A, B and C.

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