We can use a recursive definition to describe all compound propositions that use propositional variables from a specified collection. Here’s the definition for all compound propositions whose propositional variables are in \(\{p, q\}\). \[\begin{array}{ll} \textrm{Basis Step: } & p \textrm{ and } q \textrm{ are each a compound proposition} \\ \textrm{Recursive Step: } & \textrm{If } x \textrm{ is a compound proposition then so is } (\lnot x) \textrm{ and if } \\ & x \textrm{ and } y \textrm{ are both compound propositions then so is each of }\\ &(x \land y), (x \oplus y), (x \lor y), (x \to y), (x \leftrightarrow y) \end{array}\]