Let's break down the question step by step:

Given Information:

• You are working with a set $M = \mathcal{P}(\{x, y, z\})$, which is the power set of $\{x, y, z\}$. This means $M$ includes all subsets of $\{x, y, z\}$.
• The elements of $\mathcal{P}(\{x, y, z\})$ are labeled $A, B, C, \ldots, H$ as shown in the ovals.
  • $A = \{\}$ (the empty set)
  • $B = \{x\}$
  • $C = \{y\}$
  • $D = \{z\}$
  • $E = \{x, y\}$
  • $F = \{x, z\}$
  • $G = \{y, z\}$
  • $H = \{x, y, z\}$

The Relation:

• A relation $S$ between elements of $M$ is defined such that $(U, V) \in S$ if and only if $U \subseteq V$ and $U \neq V$.
  • This means that $U$ is a proper subset of $V$, excluding the case where $U = V$.

Task:

• List the members of $S$, which are ordered pairs $(U, V)$ where $U, V \in \{A, B, C, \dots, H\}$ and $U \subseteq V$, $U \neq V$.
• Additionally, draw the two-set arrow diagram for this relation.

Solution Process:

1. List all ordered pairs $(U, V)$ where $U \subseteq V$ and $U \neq V$.

   • Start by comparing each element of $M$ (labeled $A$ through $H$) to find all subsets.

2. Proper Subset Relations:

   • $A = \{\}$ is a subset of all sets except itself, so it relates to $B, C, D, E, F, G, H$.
   • $B = \{x\}$ is a subset of $E, F, H$.
   • $C = \{y\}$ is a subset of $E, G, H$.
   • $D = \{z\}$ is a subset of $F, G, H$.
   • $E = \{x, y\}$ is a subset of $H$.
   • $F = \{x, z\}$ is a subset of $H$.
   • $G = \{y, z\}$ is a subset of $H$.
   • $H = \{x, y, z\}$ is not a subset of any other set.

Final Set $S$:

Thus, the pairs are: $S = \{(A, B), (A, C), (A, D), (A, E), (A, F), (A, G), (A, H), (B, E), (B, F), (B, H), (C, E), (C, G), (C, H), (D, F), (D, G), (D, H), (E, H), (F, H), (G, H)\}$

For the arrow diagram, you would draw arrows between the elements according to these pairs, going from $U$ to $V$ (left to right).

Would you like more details on constructing the diagram or explanation of any particular step? Here are 5 related questions for further exploration:

1. How can we formally prove that $U \subseteq V$ for any given pair in $S$?
2. What is the significance of the empty set in subset relations?
3. How would the relation change if we allowed $U = V$ in the set $S$?
4. What if the set $M$ was a power set of a larger set (e.g., $\{x, y, z, w\}$)?
5. How does this type of relation relate to concepts in graph theory?

Tip: When constructing arrow diagrams, it helps to first list the elements in a systematic order to ensure no arrows are missed.