Translate each statement I-V into a propositional formula. The symbols H,L,R,S mean that Howard, Leonard, Ray or Sheldon took part in the cake-robbery.
Today is Sheldon's birthday. For this reason, Penny baked him a cake yesterday which she precautionally put into the fridge, in her apartment. Besides Sheldon and Penny, their friends Leonard, Raj and Howard have also access to the apartment (and thus can steal the birthday cake as well). And apparently, on the following day, the birthday cake is gone. Penny slept tightly and now tries to find the guilty ones. Following is certain:
I: Except for Sheldon, Leonard, Ray and Howard (and Penny who has certainly not eaten / stolen the cake), no one else has access to the apartment.
II: Howard didn't take part in the cake-robbery due to his peanut-allergy.
III: Ray would have dared to steal the cake if Sheldon supported him.
IV: If Sheldon went stealing the birthday cake, then Leonard would have supported him.
V: Leonard is too small to steal the cake from the fridge and thus, if at all, he isn't a lone perpetrator.
Text is a bit confusing for me but I hope it didn't get as confusing for you (because I translated it). There are many sub-tasks but I'm only asking for this one because if this is wrong, every other task will be wrong.. :o
So these are the formulas I created for each statement:
$\text{I: } S \wedge L \wedge R \wedge H \wedge P$
$\text{II: }\neg H$
$\text{III: } R \rightarrow (R \wedge S)$
$\text{IV: } S \rightarrow (S \wedge L)$
$\text{V: }(L \vee \neg L) \wedge (H \vee R \vee S)$
I'm not sure if this is correct? And the character Penny confused me too. But since she wasn't mentioned in the first two sentences in the task above, I think she is excluded.
For I, you want to use a disjunction, rather than a conjunction. And yes, Penny is assumed not to be involved. So, you should have $S \lor L \lor R \lor H$
II. is indeed just $\neg H$
III is a little hard to interpret, but it seems to saying that Ray would not have been involved without Sheldon's help ... which means that if Ray is involved, then Sheldon must have been involved as well. Does it also mean that if Sheldon was involved then Ray would be involved as well? It doesn't seem so: Sheldon may not have lent his support to Ray, even if Ray wanted to. So, it seems like all we have is: $R \rightarrow S$. You don't need to say $R \rightarrow (R \land S)$ (actually, this is equivalent to $R \rightarrow S$)
IV. Likewise, all you have to say here is $S \rightarrow L$
V. Here you should say that if Leonard was involved, then someone else should have been involved as well, i.e. $L \rightarrow (S \lor R \lor H)$