Let's determine the sets \( A \) and \( B \) based on their respective definitions, and then find the number of elements in each set.
1. Set A: Even numbers in \( U \)
Define \( A \) as the set of even numbers in the universal set \( U \). The even numbers within \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) are:
[tex]\[ A = \{2, 4, 6, 8, 10\} \][/tex]
2. Set B: Prime numbers in \( U \)
Define \( B \) as the set of prime numbers in the universal set \( U \). The prime numbers within \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) are:
[tex]\[ B = \{2, 3, 5, 7\} \][/tex]
Next, we find the number of elements in each set.
3. Number of elements in set A, \( n(A) \)
Count the elements in set \( A \):
[tex]\[ n(A) = 5 \][/tex]
4. Number of elements in set B, \( n(B) \)
Count the elements in set \( B \):
[tex]\[ n(B) = 4 \][/tex]
Therefore, the sets and their cardinalities are as follows:
- \( A = \{2, 4, 6, 8, 10\} \)
- \( B = \{2, 3, 5, 7\} \)
- \( n(A) = 5 \)
- [tex]\( n(B) = 4 \)[/tex]
