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Suppose that the universal set \( U \) is defined as \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

Write the sets \( A \) and \( B \) in roster notation.
[tex]\[ A = \{x \in U \mid x \text{ is an even number} \} = \{2, 4, 6, 8, 10\} \][/tex]
[tex]\[ B = \{x \in U \mid x \text{ is a prime number} \} = \{2, 3, 5, 7\} \][/tex]

Find each of the following:
[tex]\[ n(A) = \square \][/tex]
[tex]\[ n(B) = \square \][/tex]



Answer :

GinnyAnswer
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]