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Let $\alpha$ be a well-formed formula and $K(\alpha)$ be the set of sentence symbols which occur in $\alpha$. Define $K(\alpha)$ by recursion.

I need to start with a basic set, so for some $K(\alpha_{1})$, $K(\alpha_{1})=\{\emptyset, \alpha \}$. How should I proceed?

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    Not clear... If the sentence symbols are : $p_i$, then we have : *(i)* if $\alpha=p_i$, then $K(\alpha)= \{ p_i \}$. *(ii)* If $\alpha = \lnot \beta$, then $K(\alpha)=K(\beta)$, and so on... – 2017-02-08

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$K(A) = \{ A \}$ for any atomic $A$

$K(\neg \alpha) = K(\alpha)$

$K(\alpha \land \beta) = K(\alpha) \cup K(\beta)$

$K(\alpha \lor \beta) = K(\alpha) \cup K(\beta)$

... (and indeed for any two-place operator *: $K(\alpha * \beta) = K(\alpha) \cup K(\beta)$