Question

let u = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, a = {3, 5, 7, 9, 11}, b = {4, 6, 8, 10, 12}, and c = {3, 4, 6, 7, 10, 11}. list the elements of each set. (enter your answers using roster notation. enter empty or ∅ for the empty set.) (a) (a ∩ b) ∪ c (b) (a ∪ b ∪ c)ⁿ (c) (a ∩ b ∩ c)ⁿ

Explanation:

Step1: Find $A \cap B$

$A \cap B = \{3,5,7,9,11\} \cap \{4,6,8,10,12\} = \emptyset$

Step2: Find $(A \cap B) \cup C$

$(A \cap B) \cup C = \emptyset \cup \{3,4,6,7,10,11\} = \{3,4,6,7,10,11\}$

Step3: Find $A \cup B \cup C$

$A \cup B = \{3,4,5,6,7,8,9,10,11,12\}$
$A \cup B \cup C = \{3,4,5,6,7,8,9,10,11,12\} \cup \{3,4,6,7,10,11\} = \{3,4,5,6,7,8,9,10,11,12\}$

Step4: Find $(A \cup B \cup C)^c$

$(A \cup B \cup C)^c = U - (A \cup B \cup C) = \{3,4,5,6,7,8,9,10,11,12\} - \{3,4,5,6,7,8,9,10,11,12\} = \emptyset$

Step5: Find $A \cap B \cap C$

$A \cap B = \emptyset$, so $A \cap B \cap C = \emptyset \cap C = \emptyset$

Step6: Find $(A \cap B \cap C)^c$

$(A \cap B \cap C)^c = U - \emptyset = \{3,4,5,6,7,8,9,10,11,12\}$

Answer:

(a) $\{3, 4, 6, 7, 10, 11\}$
(b) $\emptyset$
(c) $\{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$