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 $S$.