The problem asks for the cardinality of the set $\{x \mid x \text{ is an integer and } x > 12 \text{ and } x < 27\}$.

Step-by-Step Solution:

1. Understand the conditions:

   • $x > 12$: $x$ must be greater than 12.
   • $x < 27$: $x$ must be less than 27.
   • $x$ must be an integer.

2. List all integers satisfying the conditions: The integers that satisfy $12 < x < 27$ are: $13, 14, 15, \dots, 26$

3. Count the integers: This is an arithmetic sequence with the first term $a = 13$ and the last term $l = 26$. The total number of terms $n$ in this sequence is given by: $n = l - a + 1$ Substituting the values: $n = 26 - 13 + 1 = 14$

Final Answer:

The cardinality of the set is 14.

Would you like further explanation or details about this?

Here are 5 related questions to expand your understanding:

1. What is the cardinality of a set and why is it important in mathematics?
2. How would the answer change if the set condition included $x \geq 12$ instead of $x > 12$?
3. Can you generalize a formula to find the number of integers between any two given numbers $a$ and $b$?
4. How would you count elements if the set were composed of non-integer values instead of integers?
5. How do sets like this appear in real-world applications?

Tip: When dealing with conditions for a set, write out the range explicitly to ensure all constraints are considered.