Equivalence of $a \rightarrow b$ and $\lnot a \vee b$ Is there a proof for the logical equivalence of $a \rightarrow b$ and $\lnot a \vee b$?
 A: Suppose we are for the moment using $p \to q$ for the informal indicative conditional, if $p$ then $q$.

  
*
  
*Suppose we are given that if $p$ then $q$ (i.e. $p \to q$). Then, since we have $\neg p \lor p$ it follows (arguing by cases) that $\neg p \lor q$.
  
*Suppose, conversely, we are given $\neg p \lor q$ is true. Then if $p$ is true that rules out the first disjunct, so we have $q$. In short, if $p$ then $q$, (i.e.  $p \to q$).
  

That pair of arguments makes it rather plausible that $p \to q$ (the indicative conditional) is inter-dervable with $\neg p \lor q$, and so is equivalent at least as far as truth-conditions go with $\neg p \lor q$.
OK: That gives us some reason to suppose that the truth-functional account of the conditional isn't just fanciful -- it gets something important right. Or so a familiar story goes.
But do note (important point!) it is a further question whether the truth-functional story about 'if' is the whole story. 
Compare, arguably '$p$ but $q$' is truth-functionally equivalent to $p \land q$: however 'but' doesn't mean just bare conjunction. 'But' does more than '$\land$' -- very roughly it signals the speaker's sense that the joint truth of $p$ and $q$ is unexpected/surprising. Well likewise, perhaps "if p then q" does more than '$\neg p \lor q$'. E.g. it arguably signals the speaker's willingness to infer $q$ should it turn out that $p$. For more on this, see e.g. http://www.logicmatters.net/resources/pdfs/Conditionals.pdf
A: I'm gonna use p and q.
Natural deduction:
\begin{align}
&1\ p \rightarrow q &&P\\
&2\ |\ \lnot (\lnot p \lor q) &&H\\
&3\ |\ |\ p &&H\\
&4\ |\ |\ q &&1,3 I \rightarrow\\
&5\ |\ |\  \lnot p \lor q &&4 I \lor\\
&6\ |\ |\  \lnot ( \lnot p \lor q) \land (\lnot p \lor q) && 2,5 I \lor \bot\\
&7\ |\ \lnot p&&3-6 I \lnot\\
&8\ |\ \lnot p \lor q&&7I\lor\\
&9\ |\ (\lnot p \lor q) \land \lnot (\lnot p \lor q)&&2,8\land \bot\\
&10\ \lnot p \lor q&&2\lnot
\end{align}
Hope this helps you!
A: Try using a truth table comparing $p\rightarrow q$ with $\lnot p \lor q$.
That's as good a proof as you need.
$p$ | $q\,$ |$\;p\rightarrow q$ | $\lnot p \lor q$
T | T |$\quad$ T $\quad$|$\quad$ T 
T | F |$\quad$ F $\quad$|$\quad$ F
F | T |$\quad$ T $\quad$|$\quad$ T
F | F |$\quad$ T $\quad$|$\quad$ T
Both expressions evaluate false if and only if $\,p\,$ is true and $\,q\,$ is false.

For developing a bit of intuition regarding the conditional (logical implication): see, e.g.,
$(1)$ this post, and
$(2)$ this post.
A: Think of it this way. Suppose I signed a contract to you, saying "If I win the lottery, then I will give you $\$1$ billion." The only situation in which my statement would be false (contract violated) is: I win the lottery and don't give you the money I promised. Otherwise, I'm just fine. In particular, if I never win the lottery, you have no claim on any of my money. On the other hand, I can give you the money just for kicks, and I still haven't violated the contract.
Similarly, what if I said "If it's raining, I carry an umbrella." You couldn't call me a liar if it isn't raining, nor if I have an umbrella (regardless of the weather).
A general conditional statement "if (hypothesis) then (conclusion)" is (by this reasoning) the same as saying "(hypothesis) is false or (conclusion) is true."
