TITLE: My professor says this is NOT a typo, but this does not appear to be logically valid.
QUESTION [8 upvotes]: $$\begin{array}{rlll}
 1. & \sim H \lor \sim G & \text{Premise} & \\
 2. & H\land (G\lor H)     & \text{Premise} & \text{DEDUCE $F\& H$}
 \end{array}$$

Using the rules of inference from standard propositional logic, we have to deduce $F \land H$.
Well, if we can deduce $F \land H$, we can also deduce $F$ through simplification. That means that these two premises can imply any statement, $F$, ever.
It does not appear possible to deduce the conclusion.
I feel like the only thing we can deduce is $H \land \sim G$. I even thought of what appears to be a counterexample. Let

$H$: I live at home.

$G$: I own a piece of gold.

$F$: I own a Ferrari.


Premise 1 translates to "It is not the case that I live at home or it is not the case that I own a piece of gold." which is true. Premise 2 translates to "I live at home and either I own a piece of gold or I live at home." which is also true.
However, the conclusion $F \land H$ translates to "I own a Ferrari and I live at home", which is false.
(original scan of problem here)

REPLY [3 votes]: I suspect that Premise 1 is meant to be:
$  \lnot ( H \lor \lnot G ) $
which simplifies to:
$ \lnot H \land G $
and from which it follows, by and-ing to the simplification of Premise 2 as simply H:
$ H \land \lnot H  \land G $
and then simply:
$ False $
From a falsehood, one can of course imply everything.