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Let U={1,2,3,4,5,6,7,8,9,10}, A={1,3,...

Let U={1,2,3,4,5,6,7,8,9,10},
A={1,3,5},B={2,4,6},C={4,5,6}.
Find (i) `A^(c)capB^(c)`
(ii) `(AuuB)^(c)capC^(c)`.

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To solve the problem step by step, we will find the required sets as per the given question. ### Given: - Universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) - Set \( A = \{1, 3, 5\} \) - Set \( B = \{2, 4, 6\} \) - Set \( C = \{4, 5, 6\} \) ### (i) Find \( A^c \cap B^c \) **Step 1: Find \( A^c \)** The complement of set \( A \) (denoted as \( A^c \)) includes all elements in the universal set \( U \) that are not in \( A \). \[ A^c = U - A = \{2, 4, 6, 7, 8, 9, 10\} \] **Step 2: Find \( B^c \)** The complement of set \( B \) (denoted as \( B^c \)) includes all elements in the universal set \( U \) that are not in \( B \). \[ B^c = U - B = \{1, 3, 5, 7, 8, 9, 10\} \] **Step 3: Find \( A^c \cap B^c \)** Now, we find the intersection of \( A^c \) and \( B^c \), which includes elements that are common to both sets. \[ A^c \cap B^c = \{2, 4, 6, 7, 8, 9, 10\} \cap \{1, 3, 5, 7, 8, 9, 10\} \] The common elements are \( \{7, 8, 9, 10\} \). Thus, \[ A^c \cap B^c = \{7, 8, 9, 10\} \] ### (ii) Find \( (A \cup B)^c \cap C^c \) **Step 1: Find \( A \cup B \)** The union of sets \( A \) and \( B \) (denoted as \( A \cup B \)) includes all distinct elements from both sets. \[ A \cup B = \{1, 3, 5\} \cup \{2, 4, 6\} = \{1, 2, 3, 4, 5, 6\} \] **Step 2: Find \( (A \cup B)^c \)** The complement of \( A \cup B \) (denoted as \( (A \cup B)^c \)) includes all elements in \( U \) that are not in \( A \cup B \). \[ (A \cup B)^c = U - (A \cup B) = \{7, 8, 9, 10\} \] **Step 3: Find \( C^c \)** The complement of set \( C \) (denoted as \( C^c \)) includes all elements in \( U \) that are not in \( C \). \[ C^c = U - C = \{1, 2, 3, 7, 8, 9, 10\} \] **Step 4: Find \( (A \cup B)^c \cap C^c \)** Now, we find the intersection of \( (A \cup B)^c \) and \( C^c \). \[ (A \cup B)^c \cap C^c = \{7, 8, 9, 10\} \cap \{1, 2, 3, 7, 8, 9, 10\} \] The common elements are \( \{7, 8, 9, 10\} \). Thus, \[ (A \cup B)^c \cap C^c = \{7, 8, 9, 10\} \] ### Final Answers: (i) \( A^c \cap B^c = \{7, 8, 9, 10\} \) (ii) \( (A \cup B)^c \cap C^c = \{7, 8, 9, 10\} \)
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