In set theory, the notation $|A \cap B \text{ only}|$ typically represents the number of elements that are in both sets $A$ and $B$, but not in $C$ (if $C$ is involved). This can be written mathematically as:

$|A \cap B \text{ only}| = |(A \cap B) - (A \cap B \cap C)|$

On the other hand, $|A \cap B / C|$ usually represents the number of elements in the set $A \cap B$ that are not in $C$, which is:

$|A \cap B / C| = |(A \cap B) - C|$

However, this notation can sometimes be ambiguous and context-dependent. To clarify, let's establish the precise relationships using standard set theory notation.

Notations and Meanings:

1. $|A \cap B \text{ only}|$ means $|A \cap B - (A \cap B \cap C)|$.
2. $|A \cap B / C|$ means $|A \cap B - C|$, which simplifies to $|A \cap B - (A \cap B \cap C)|$.

Thus, if we assume that $|A \cap B / C|$ and $|A \cap B \text{ only}|$ are intended to mean the same thing, then their value would be:

$|A \cap B| - |A \cap B \cap C|$

This formula indicates the number of elements that are in both $A$ and $B$, but not in $C$.

Therefore, $|A \cap B \text{ only}|$ and $|A \cap B / C|$ do indeed represent the same quantity when interpreted as above.

Would you like more details on this topic, or do you have any other questions?

Here are 5 questions you might ask next:

1. How do you find the union of multiple sets in set theory?
2. What is the principle of inclusion-exclusion in set theory?
3. Can you explain De Morgan's laws with examples?
4. How do you prove that two sets are equal?
5. What is a Venn diagram, and how is it used in set theory?

Tip: Always clearly define your sets and their relationships when working with set theory problems to avoid ambiguity and ensure accurate results.