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We are tasked with finding the cardinality of the set \((A - C) \cap B\), given the following sets:

- \( A = \{x \mid x \in \mathbb{N}\} \) (the set of all natural numbers),
- \( B = \{x \mid -5 < x < 105, x \in \mathbb{R}\} \) (all real numbers between -5 and 105),
- \( C = \{x \mid x \text{ is a rational number}, 10 < x \leq 80\} \) (rational numbers between 10 and 80, inclusive of 80 but not 10).

### Steps:

1. **Interpretation of Sets:**
   - \(A\) consists of all natural numbers, i.e., \( A = \{1, 2, 3, \dots\}\).
   - \(B\) consists of all real numbers between \(-5\) and \(105\), i.e., \( B = (-5, 105) \subseteq \mathbb{R}\).
   - \(C\) consists of all rational numbers in the interval \( (10, 80] \).

2. **Set Operations:**
   - **\(A - C\):** This represents the set of natural numbers that are **not in** the set \(C\). Since \(C\) contains rational numbers between 10 and 80, we exclude the natural numbers between 11 and 80 from \(A\). So, \(A - C = \{1, 2, 3, \dots, 10, 81, 82, \dots\}\).
   - **Intersection with \(B\):** Now, we intersect \(A - C\) with \(B\). Since \(B\) includes all real numbers between \(-5\) and \(105\), the intersection will include all natural numbers in this range, but limited to those that are in \(A - C\).

   Thus, \( (A - C) \cap B = \{1, 2, 3, \dots, 10, 81, 82, \dots, 104\} \).

3. **Cardinality of \( (A - C) \cap B \):**
   - The numbers in this set are the natural numbers from 1 to 10 and from 81 to 104.
   - The count of natural numbers from 1 to 10 is 10.
   - The count of natural numbers from 81 to 104 is \(104 - 81 + 1 = 24\).
   - Therefore, the cardinality is \(10 + 24 = 34\).

### Answer:
The cardinality of the set \((A - C) \cap B\) is **34**.

Would you like further details or have any other questions?

Here are some related questions you might find useful:
1. What is the difference between natural numbers and rational numbers?
2. How would the result change if the set \(C\) included irrational numbers?
3. What happens if we change the interval for \(B\) to a larger range?
4. How do we determine cardinality for infinite sets?
5. Can we generalize this concept for other types of sets, such as integers?

**Tip:** When dealing with set operations, it's useful to visualize intervals on the number line to avoid confusion.

Consider the following sets: A = {x | x ∈ N}, B = {x | -5 < x < 105, x ∈ R}, C = {x | x is a rational number, 10 < x ≤ 80}. Find the cardinality of the set (A - C) ∩ B.

Solve a set theory problem involving natural numbers, real numbers, and rational numbers. Learn how to find the cardinality of the set (A - C) ∩ B using set difference and intersection operations. Suitable for students in Grades 9-12.

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