Independent set of formulas of the sentential logic A set $\Gamma$ of well formed formulas of the sentential logic is called independent if for each $\varphi\in\Gamma$, $\Gamma-\{\varphi\}\nRightarrow\varphi\\$.
1 when $\Gamma=\{\varphi\}$ is independent?
2 Is $\{A\rightarrow B, B\rightarrow C, C\rightarrow A\}$ independent? ($A,B,C$ are sentencial letters).
In the first case I guess that $\varphi$ should be a tautology but I get confused because $\emptyset\Rightarrow\varphi$ has a meaning?
And for the second I said no, because I can easily find valuation that make two of them true but not the third one.
If it is useful $\alpha\Rightarrow\beta$ holds iff $\alpha\rightarrow\beta$ is a tautology. 

You haven't said whether $\implies$ is semantic or syntactic entailment, but the same goes either way. I'll assume you mean semantic entailment, but you can easily adjust the answer if you meant syntactic entailment.


*

*$\{\varphi\}$ is independent [on your definition] iff $\emptyset \nvDash \varphi$,  i.e. iff $\varphi$ is not a tautology. But why does $\emptyset \vDash \varphi$ say that $\varphi$ is a tautology? Recall: $\Delta \vDash \varphi$ says that any valuation which makes $\varphi$ false must make some wff in $\Delta$ false. So: $\emptyset \vDash \varphi$ says that any valuation which makes $\varphi$ false must make some wff in $\emptyset$ false. But there are no wffs in the empty set to make false, so that's equivalent to saying no valuation makes $\varphi$ false, i.e. $\varphi$ is a tautology.

*Yes, except you need three valuations since you need to consider the three different cases where you extract in turn one of the propositions from the given set and see if the extracted set follows from the remainder. 
