Let's go through each set and classify them as finite (F) or infinite (I) and determine their cardinality (the number of elements).

1. A = set of odd prime numbers

   • Classification: Infinite (I)
   • Cardinality: Infinite
     There are infinitely many prime numbers, and the odd ones (e.g., 3, 5, 7, 11, etc.) form an infinite set.

2. A = {factors of 12}

   • Classification: Finite (F)
   • Cardinality: 6
     The factors of 12 are {1, 2, 3, 4, 6, 12}, so the set has 6 elements.

3. A = {counting numbers between 1 and 2}

   • Classification: Finite (F)
   • Cardinality: 0
     There are no counting numbers strictly between 1 and 2, so the set is empty, and the cardinality is 0.

4. A = {x | x is a counting number and x + 3 = 4}

   • Classification: Finite (F)
   • Cardinality: 1
     Solving the equation $x + 3 = 4$ gives $x = 1$, so the set contains only one element, {1}.

5. A = set of letters in the word ‘PHILIPPINES’

   • Classification: Finite (F)
   • Cardinality: 7
     The distinct letters in "PHILIPPINES" are {P, H, I, L, N, E, S}, which gives 7 unique elements.

6. A = set of letters in the word ‘INFINITE’

   • Classification: Finite (F)
   • Cardinality: 5
     The distinct letters in "INFINITE" are {I, N, F, T, E}, which gives 5 unique elements.

7. A = set of PSHS campuses

   • Classification: Finite (F)
   • Cardinality: 16
     As of the latest information, there are 16 Philippine Science High School (PSHS) campuses.

8. A = {core values of Pisay}

   • Classification: Finite (F)
   • Cardinality: 4
     The core values of PSHS are typically Integrity, Excellence, Service, and Resilience, so the set has 4 elements.

9. A = set of natural numbers greater than 1 but less than 5

   • Classification: Finite (F)
   • Cardinality: 3
     The natural numbers between 1 and 5 are {2, 3, 4}, so the set has 3 elements.

10. A = {5, 10, 15, 20, 25, ...}

    • Classification: Infinite (I)
    • Cardinality: Infinite
      This is the set of multiples of 5, and it is an infinite sequence.

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Let me know if you want any additional details or explanations. Here are some follow-up questions:

1. What is the cardinality of the set of even prime numbers?
2. Can you give an example of a finite set with a cardinality of 10?
3. Is the set of real numbers between 0 and 1 finite or infinite?
4. What is the difference between finite and infinite sets?
5. How do you find the cardinality of a set with repeating elements like in words?

Tip: When dealing with finite sets, the cardinality is simply the number of unique elements in the set.