Question

Let \( A = \{x : x = 4n + 1, n \in \mathbb{Z}, 0 \leq n < 4 \} \)
Let \( B = \{x : x = 15n + 4, n \in \mathbb{N}, n \leq 3 \} \)
Let \( C = \{x : x \text{ is a prime number}, x \in A \cup B \} \)
Then the cardinal number of set \( C \) is

• A. \( \emptyset \)
• B. 3
• C. 4
• D. 5

Hint:

To find the cardinal number of a set, identify the elements of the set and count them. In this case, count the prime numbers in the union of sets \( A \) and \( B \).

Answer 1

The Correct Option is B

Step 1: Identify the elements of set A.
The set \( A \) is given by:
\[ A = \{x : x = 4n + 1, n \in \mathbb{Z}, 0 \leq n < 4 \} \] Substitute \( n = 0, 1, 2, 3 \) into \( x = 4n + 1 \):
\[ A = \{1, 5, 9, 13\} \]

Step 2: Identify the elements of set B.
The set \( B \) is given by:
\[ B = \{x : x = 15n + 4, n \in \mathbb{N}, n \leq 3 \} \] Substitute \( n = 0, 1, 2, 3 \) into \( x = 15n + 4 \):
\[ B = \{4, 19, 34, 49\} \]

Step 3: Find the union of A and B.
The union of \( A \) and \( B \) is:
\[ A \cup B = \{1, 5, 9, 13, 4, 19, 34, 49\} \]

Step 4: Find the prime numbers in \( A \cup B \).
The prime numbers in \( A \cup B \) are:
\[ \{1, 5, 9, 13, 4, 19, 34, 49\} \quad \text{(Prime numbers are 5, 13, and 19)} \]

Step 5: Cardinal number of set \( C \).
The set \( C \) contains the prime numbers in \( A \cup B \), so the cardinal number of \( C \) is 3.

Step 6: Final Answer.
The correct answer is option (B), as the cardinal number of set \( C \) is 3.