https://file.aiquickdraw.com/mathbot/file/1729735527569xqtjtb7h.jpg

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.

The set M = P({x,y,z}). The elements of P({x,y,z}) are labelled A,...,H as indicated. The relation S with domain M and codomain M is defined by the rule: for any U, V ∈ M, (U, V) ∈ S if and only if U ⊆ V and U ≠ V. List the members in S.

Explore how to list all proper subset relations for the power set of {x, y, z}, including ordered pairs and arrow diagrams. This problem involves set theory concepts and relations, suitable for undergraduate discrete mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation