Exercise 11, § 4

Let A be a set and let $\mathfrak{R}$ be a subset of the set $\mathfrak{F}(A)$ of finite subsets of A. $\mathfrak{R}$ is said to be mobile if it satisfies the following condition :

(MO) If X, Y are two distinct elements of $\mathfrak{R}$ and if $z \in X \cap Y$, then there exists $Z \subset X \cap Y$ belonging to $\mathfrak{R}$ such that $z \notin Z$.

A subset P of A is then said to be pure if it contains no set belonging to $\mathfrak{R}$.

(a) Show that every pure subset of A is contained in a maximal pure subset of A.

(b) Let M be a maximal pure subset of A. Show that for each $x \in \complement M$ there exists a unique finite subset $E_M(x)$ of M such that $E_M(x) \cup \{x\} \in \mathfrak{R}$. Moreover, if $y \in E_M(x)$, the set $(M \cup \{x\}) - \{y\}$ is a maximal pure subset of A.

(c) Let M, N be two maximal pure subsets of A, such that $N \cap \complement M$ is finite. Show that Card (M) = Card (N). (Proof by induction on the cardinal of $N \cap \complement M$, using (b).)

(d) Let M, N be two maximal pure subsets of A, and put $N' = N \cap \complement M$, $M' = M \cap \complement N$. Show that

\* Deduce that Card (M) = Card (N) (by virtue of (c), we are reduced to the case where N' and M' are infinite; show then that Card (M') $\leqslant$ Card (N')). \*