To solve the problem, we need to find the complement of the union of two sets, \( P \) and \( Q \), within the universal set \( U \). Let's break this down step-by-step. ### Step 1: Define the Universal Set \( U \) The universal set \( U \) is defined as the set of even numbers between 0 and 30. This includes: \[ U = \{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\} \] ### Step 2: Define the Set \( P \) Set \( P \) consists of multiples of 6 between 0 and 30. The multiples of 6 in this range are: \[ P = \{0, 6, 12, 18, 24, 30\} \] ### Step 3: Define the Set \( Q \) Set \( Q \) consists of multiples of 4 between 0 and 30. The multiples of 4 in this range are: \[ Q = \{0, 4, 8, 12, 16, 20, 24, 28\} \] ### Step 4: Find the Union \( P \cup Q \) The union of sets \( P \) and \( Q \) includes all elements that are in either set. We combine the elements from both sets: \[ P \cup Q = \{0, 6, 12, 18, 24, 30\} \cup \{0, 4, 8, 12, 16, 20, 24, 28\} \] When we combine these, we list each unique element: \[ P \cup Q = \{0, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30\} \] ### Step 5: Find the Complement \( (P \cup Q)^c \) The complement of \( P \cup Q \) within the universal set \( U \) consists of all elements in \( U \) that are not in \( P \cup Q \). We will subtract the elements of \( P \cup Q \) from \( U \): \[ (P \cup Q)^c = U - (P \cup Q) \] This means we take each element in \( U \) and check if it is in \( P \cup Q \): - From \( U \): \( 0 \) (in \( P \cup Q \)), \( 2 \) (not in \( P \cup Q \)), \( 4 \) (in \( P \cup Q \)), \( 6 \) (in \( P \cup Q \)), \( 8 \) (in \( P \cup Q \)), \( 10 \) (not in \( P \cup Q \)), \( 12 \) (in \( P \cup Q \)), \( 14 \) (not in \( P \cup Q \)), \( 16 \) (in \( P \cup Q \)), \( 18 \) (in \( P \cup Q \)), \( 20 \) (in \( P \cup Q \)), \( 22 \) (not in \( P \cup Q \)), \( 24 \) (in \( P \cup Q \)), \( 26 \) (not in \( P \cup Q \)), \( 28 \) (in \( P \cup Q \)), \( 30 \) (in \( P \cup Q \)). Thus, the elements that are not in \( P \cup Q \) are: \[ (P \cup Q)^c = \{2, 10, 14, 22, 26\} \] ### Step 6: Identify the Correct Option Now, we compare our result \( (P \cup Q)^c = \{2, 10, 14, 22, 26\} \) with the provided options: - A. \( \{2, 10, 14, 22, 26\} \) - **Correct** - B. \( \{0, 10, 14, 22, 26\} \) - Incorrect (0 is in \( P \cup Q \)) - C. \( \{2, 4, 14, 18, 26\} \) - Incorrect (4 and 18 are in \( P \cup Q \)) - D. \( \{0, 2, 6, 22, 26\} \) - Incorrect (0 and 6 are in \( P \cup Q \)) ### Summary - The universal set \( U \) consists of even numbers between 0 and 30. - The union \( P \cup Q \) includes all multiples of 6 and 4 within that range. - The complement \( (P \cup Q)^c \) consists of even numbers not included in \( P \cup Q \). - The correct answer is option A: \( \{2, 10, 14, 22, 26\} \). ### Revision Summary - Understand the definitions of universal set, union, and complement. - Carefully list elements in each set and their union. - Subtract elements of the union from the universal set to find the complement. - Verify your answer against the provided options to ensure accuracy.