Two propositions concerning refinements of partitions

Question

I am having problems understanding parts of Richard Zuber's Set Partitions and the Meaning of the Same (published online by the Journal of Logic, Language and Information, 2016, doi:10.1007/s10849-016-9241-z).

I will first give some relevant definitions and then explain what I don't understand.

• Given a mapping $f$ on $A \subseteq E$ (for universe $E$), consider the equivalence relation $e_f$ on $A$ defined as $\langle x, y \rangle \in e_f$ iff $f(x) = f(y)$.

• Let $R$ be a binary relation in the universe $E$. Where $xR$ denotes the set $\{\ x' : \langle x, x' \rangle \in R\}$, define the following two mappings of $E$:

  1. $f_1(x) = xR$ and
  2. $f_2(x) = |xR|$

  With $R$ associate the following two equivalence relations:

  1. $e_R = \{\ \langle x, y \rangle : xR = yR \}$

  2. $e_{R,n} = \{\ \langle x, y \rangle : |xR| = |yR| \}$

• If $R$ is a binary relation, let $\Pi_R(X)$ denote the partition of the set $X$ defined by the relation $e_R$, and let $\Pi_{\thinspace R,\thinspace n}\thinspace(X)$ denote the partition of the set $X$ defined by the relation $e_{R, n}$.

Now on p.7, paragraph 3 of the article is the passage I do not understand. Zuber writes:

Partitions (of the same set) can be partially ordered by the refinement relation: the partition $\Pi_1(X)$ refines the partition $\Pi_2(X)$ iff for any block $B_1 \in \Pi_1(X)$ there exists a block $B_2 \in \Pi_2(X)$ such that $B_1 \subseteq B_2$. It is easy to see that $\Pi_R(X)$ refines $\Pi_{\thinspace R,\thinspace n}\thinspace(X)$. Moreover we also obtain a refinement relation on some of the above partitions, if we consider relations $R_C$ and $R_D$ with $C$ and $D$ ordered by inclusion. More specifically we have:

Proposition 1: Let $C ⊆ D$. Then for any $X \neq \emptyset$, any $R \neq \emptyset$, any $S \neq \emptyset$: $\Pi_{R_D} (X)$ refines $\Pi_{R_C} (X)$

Given that $aR = bR$ iff $aR' = bR'$, $aR = aS$ iff $aR' = aS'$ and $|aR| = |aS|$ iff $|aR'| = |aS'|$, for $R$ and $S$ binary, we also have:

Proposition 2: (i) $\Pi_R(X) = \Pi_{R\thinspace'}(X)$
(ii) $\Pi_{\thinspace R,\thinspace n}\thinspace(X) = \Pi_{\thinspace R\thinspace',\thinspace n}\thinspace(X)$

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Question number 1: Is there an error in the statement of Proposition 1? For if $C ⊆ D$ shouldn't it be the case that $\Pi_{R_C} (X)$ refines $\Pi_{R_D} (X)$, given that $C \subseteq D$ and not, as written in Proposition 1, that $\Pi_{R_D}$ refines $\Pi_{R_C}$?

Question number 2: How does Proposition 2 follow from assuming (1)-(3), as Zuber seems to be suggesting?

1. $aR = bR$ iff $aR' = bR'$

2. $aR = aS$ iff $aR' = aS'$

3. $|aR| = |aS|$ iff $|aR'| = |aS'|$