In https://encyclopediaofmath.org/wiki/Characteristic_function_of_a_set, various properties of characteristic functions are discussed.
However, the notation used there is confusing. In particular how is $\chi_E (1 - \chi_F )$ to be read? If $\chi_E$ is the characteristic function of $E \subseteq X$ and $(1 - \chi_F )$ outputs a number, then $\chi_E$ might not be able to apply to the output of $(1 - \chi_F )$, unless $E$ is assumed to contain $1$ and $0$.
Likewise, what is the implicit ordering relative to which $sup_{\alpha}$ and $inf_{\alpha}$ are to be understood? (i.e, they are the supremum and infimum with respect to what partial ordering?).