Proposition 17

In order that a nonempty set $\Phi$ of subsets of a set $A$ be a clan, it is necessary and sufficient that it satisfy the following condition:

(CL) For every pair of sets $M, N$ belonging to $\Phi$, the sets $M \cup N$ and $M \cap \mathbf{C}N$ belong to $\Phi$.¹

The condition is necessary, by virtue of the relations

To show that it is sufficient, we first observe that it implies that, for any two sets $M, N$ in $\Phi$, $M \cap N$ belongs to $\Phi$ since $M \cap N = M \cap \mathbf{C}(M \cap \mathbf{C}N)$. Let $\mathscr{E}(\Phi)$ be the set of linear combinations, with real coefficients, of the characteristic functions of the sets of $\Phi$. Since $\varphi_M \varphi_N = \varphi_{M \cap N}$, $\mathscr{E}(\Phi)$ is an algebra. Everything comes down to showing that if $M$ is a subset of $A$ such that $\varphi_M = \sum_i c_i \varphi_{M_i}$, where the $M_i$ belong to $\Phi$, then $M \in \Phi$. This will result from the following lemma: