Questions: For which of the following is it true that n(A)+n(B)=n(A ∪ B) ? a. A=a, b, c, B=c, d b. A=a, b, c, B=b, e c. A=a, b, c, B=∅ a. The statement n(A)+n(B)=n(A ∪ B) is . b. The statement n(A)+n(B)=n(A ∪ B) is c. The statement n(A)+n(B)=n(A ∪ B) is

Solution Steps

To determine if \( n(A) + n(B) = n(A \cup B) \) is true for each pair of sets \( A \) and \( B \), we need to check if the sets \( A \) and \( B \) are disjoint (i.e., they have no elements in common). If they are disjoint, then the equation holds true.

1. For each pair of sets \( A \) and \( B \), find the union \( A \cup B \).
2. Calculate the number of elements in \( A \), \( B \), and \( A \cup B \).
3. Check if \( n(A) + n(B) = n(A \cup B) \).

For the first pair of sets \( A = \{a, b, c\} \) and \( B = \{c, d\} \):

• The union \( A \cup B = \{a, b, c, d\} \).
• We find \( n(A) = 3 \), \( n(B) = 2 \), and \( n(A \cup B) = 4 \).
• Thus, \( n(A) + n(B) = 3 + 2 = 5 \) which is not equal to \( n(A \cup B) = 4 \).
• Therefore, the statement \( n(A) + n(B) = n(A \cup B) \) is False.

For the second pair of sets \( A = \{a, b, c\} \) and \( B = \{b, e\} \):

• The union \( A \cup B = \{a, b, c, e\} \).
• We find \( n(A) = 3 \), \( n(B) = 2 \), and \( n(A \cup B) = 4 \).
• Thus, \( n(A) + n(B) = 3 + 2 = 5 \) which is not equal to \( n(A \cup B) = 4 \).
• Therefore, the statement \( n(A) + n(B) = n(A \cup B) \) is False.

For the third pair of sets \( A = \{a, b, c\} \) and \( B = \varnothing \):

• The union \( A \cup B = \{a, b, c\} \).
• We find \( n(A) = 3 \), \( n(B) = 0 \), and \( n(A \cup B) = 3 \).
• Thus, \( n(A) + n(B) = 3 + 0 = 3 \) which is equal to \( n(A \cup B) = 3 \).
• Therefore, the statement \( n(A) + n(B) = n(A \cup B) \) is True.

Final Answer