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.