The given expression in words is:

"M is defined as the set of all elements $x$ such that there exists some set $M$ in $S$ for which $x$ is an element of $M$."

Here’s a breakdown of the notation:

• M := This defines the set $M$.
• { x | ... } This is set-builder notation, meaning $M$ consists of elements $x$ that satisfy the condition inside.
• exists $M \in S$ This means there is some set $M$ that belongs to the set $S$.
• $x \in M$ This means $x$ is an element of $M$.

In summary:

Set $M$ contains all elements $x$ that belong to at least one of the sets contained in the set $S$.

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Would you like further details or clarification on set theory or this expression?

Here are five related questions:

1. What is the difference between set membership and subset notation?
2. How does set-builder notation differ from listing elements directly?
3. Can the set $S$ contain sets of different types (e.g., finite and infinite sets)?
4. What are typical applications of set-builder notation in mathematics?
5. How do existential quantifiers differ from universal quantifiers in logic?

Tip: Always carefully distinguish between sets and their elements when interpreting set-theoretic expressions.