Proving $(A \land B) \to C$ and $A \to (B \to C)$ are equivalent 
Prove that $(A \land B) \rightarrow C$ is equivalent to $A \rightarrow (B \rightarrow C)$ in two ways: by semantics and syntax.

Can somebody give hints or answer to solve it?
 A: Semantically it is easy. You can do a truth-table, and need to show the two wffs agree in truth-value for every valuation of $A$, $B$ and $C$.
Syntactically it will depend on the proof system you are being asked to use. But in a natural deduction system:

Take $A \land B \to C$ as a premiss. Suppose (temporary assumption) $A$. Now suppose (temporary assumption) $B$. Those two assumptions give you $A\ \land B$. Modus ponens using the original premiss gives you $C$. Now you need to discharge the two assumptions in turn to derive first $B \to C$ (with $A$ still assumed), and then $A \to (B \to C)$ (on no assumptions beyond the original premiss).
Take $A \to (B \to C)$ as a premiss. Suppose (temporary assumption) $A \land B$. Extract the conjuncts and two applications of modus ponens using the original premiss gives you $C$. Now you need to discharge the assumptions to derive  $A \land B \to C$.

Your task is now to write up those proof-sketches according to your favourite way of laying out ND proofs (Fitch-style or Gentzen-style). 
[PS If you are being forced to use an axiomatic Hilbert-style system, complain bitterly to your instructor. If your syntactic system is a tree-system as in my book then you first assume $A \to (B \to C)$ and $\neg(A \land B \to C)$ and get your tree to close (just automatic) and then ...well, you should know how the story goes!]
A: Here is a third (also syntactic) way, to supplement Peter's two good solutions.  
$$(A\wedge B)\to C$$
$$\neg(A \wedge B) \vee C$$
$$(\neg A \vee \neg B) \vee C$$
$$\neg A \vee (\neg B \vee C)$$
$$\neg A \vee (B\to C)$$
$$A\to (B\to C)$$
A: Semantically you can just consider two cases.  1) Suppose A is true, and 2) Suppose A is false.  Since all atomic propositions in classical logic are either true or false, but not both, this method will work.
Syntactically, we'll need to know the proof system (the axioms and the rules of inference for your system) to know how to solve this.
