make an example with M := { x | exists M ∈ S: x ∈ M }

In set theory, the expression \( M := \{ x \mid \exists M \in S : x \in M \} \) defines a set \( M \) based on a condition involving elements and another set \( S \).

Let's break it down and construct an example step by step:

### General Structure
- \( M := \{ x \mid \exists M \in S : x \in M \} \)
  
This means that \( M \) is the set of elements \( x \) such that there exists a set \( M \) in a collection of sets \( S \) where \( x \) is an element of \( M \). 

### Example

Let’s define a specific example where the collection \( S \) is a set of sets:

- Let \( S = \{ A, B \} \), where:
  - \( A = \{ 1, 2, 3 \} \)
  - \( B = \{ 3, 4, 5 \} \)

Now, applying the original expression:

\[ M := \{ x \mid \exists M \in S : x \in M \} \]

This means that \( M \) contains elements \( x \) such that \( x \) is in either set \( A \) or set \( B \) (since \( A \) and \( B \) are the only members of \( S \)).

Thus:

- \( x = 1 \in A \), so \( 1 \in M \)
- \( x = 2 \in A \), so \( 2 \in M \)
- \( x = 3 \in A \) and \( x = 3 \in B \), so \( 3 \in M \)
- \( x = 4 \in B \), so \( 4 \in M \)
- \( x = 5 \in B \), so \( 5 \in M \)

The resulting set \( M \) is the union of the sets \( A \) and \( B \):

\[
M = \{ 1, 2, 3, 4, 5 \}
\]

### Interpretation
The set \( M \) is constructed by collecting all elements that belong to at least one of the sets in \( S \). In this case, it's simply the union of the sets \( A \) and \( B \).

### Summary
In this example:
- \( S = \{ \{ 1, 2, 3 \}, \{ 3, 4, 5 \} \} \)
- \( M := \{ x \mid \exists M \in S : x \in M \} = \{ 1, 2, 3, 4, 5 \} \)

Would you like more details or have any questions?

Here are five follow-up questions:
1. How would the result change if the sets in \( S \) had overlapping elements?
2. Can you construct a scenario where the resulting set \( M \) is empty?
3. How does this relate to the concept of unions in set theory?
4. What would happen if \( S \) contained an infinite number of sets?
5. How does this set-building notation differ from typical set-builder notation?

**Tip:** In set theory, the expression \( \exists M \in S : x \in M \) often leads to the union of sets within the collection \( S \).

Explore set theory with the expression M := { x | exists M ∈ S: x ∈ M }, and learn how to construct sets based on conditions. This detailed guide covers the union of sets and set-builder notation, suitable for students in Grades 9-12.

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