Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{3, 4, 5, 6, 7\} \) be two sets in the universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \). Find the complement of the set \( A \bigcup B \) in \( U \).
Answer
Step 3: Subtracting the elements of \( A \bigcup B \) from the universal set \( U \), we get \( U - (A \bigcup B) = \emptyset \), which is the empty set because all elements of \( A \bigcup B \) are in the universal set \( U \).
Step 1 ：Step 1: Find the union of sets \( A \) and \( B \), denoted as \( A \bigcup B \). The union of two sets is the set of elements which are in \( A \), in \( B \), or in both \( A \) and \( B \). So, \( A \bigcup B = \{1, 2, 3, 4, 5, 6, 7\} \).
Step 2 ：Step 2: The complement of a set \( A \) in a universal set \( U \), denoted by \( A' \) or \( \overline{A} \), is the set of elements in \( U \) but not in \( A \). Therefore, the complement of \( A \bigcup B \) in \( U \) is \( U - (A \bigcup B) \).
Step 3 ：Step 3: Subtracting the elements of \( A \bigcup B \) from the universal set \( U \), we get \( U - (A \bigcup B) = \emptyset \), which is the empty set because all elements of \( A \bigcup B \) are in the universal set \( U \).
