Let $ A $ be a set. Given a subset $ S $ of $ A $ , we obtain a function $ \chi_S{\colon}\linebreak[0] A \to 2 $ (the characteristic function of $ S \subseteq A $ ), where $ \chi_S(a) = \begin{cases} \true &\text{if }a \in S, \ \false &\text{if }a \not\in S \end{cases} $ ( $ a \in A $ ). Conversely, given a function $ f{\colon}\linebreak[0] A \to 2 $ , we obtain a subset $ f^{ - 1}{\true } = { a \in A \mid f(a) = \true } $ of $ A $ .