Question

let ( u = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ), ( a = {1, 3, 5, 7, 9} ), ( b = {2, 4, 6, 8, 10} ), and ( c = {1, 2, 4, 5, 8, 9} ). list the elements of each set. (enter your answers using roster notation. enter empty or ( varnothing ) for the empty set.)
(a) ( a^c cap (b cap c^c) )
(b) ( (a cup b^c) cup (b cap c^c) )
(c) ( (a cup b)^c cap c^c )

Explanation:

Step1: Find complements

$A^c = U - A = \{2,4,6,8,10\}$
$B^c = U - B = \{1,3,5,7,9\}$
$C^c = U - C = \{3,6,7,10\}$

Step2: Solve part (a): $A^c \cap (B \cap C^c)$

First find $B \cap C^c$:
$B \cap C^c = \{2,4,6,8,10\} \cap \{3,6,7,10\} = \{6,10\}$
Then find intersection with $A^c$:
$A^c \cap (B \cap C^c) = \{2,4,6,8,10\} \cap \{6,10\} = \{6,10\}$

Step3: Solve part (b): $(A \cup B^c) \cup (B \cap C^c)$

First find $A \cup B^c$:
$A \cup B^c = \{1,3,5,7,9\} \cup \{1,3,5,7,9\} = \{1,3,5,7,9\}$
We already know $B \cap C^c = \{6,10\}$ from Step2.
Now find the union:
$(A \cup B^c) \cup (B \cap C^c) = \{1,3,5,7,9\} \cup \{6,10\} = \{1,3,5,6,7,9,10\}$

Step4: Solve part (c): $(A \cup B)^c \cap C^c$

First find $A \cup B$:
$A \cup B = \{1,3,5,7,9\} \cup \{2,4,6,8,10\} = \{1,2,3,4,5,6,7,8,9,10\}$
Then find its complement:
$(A \cup B)^c = U - (A \cup B) = \emptyset$
Now find intersection with $C^c$:
$(A \cup B)^c \cap C^c = \emptyset \cap \{3,6,7,10\} = \emptyset$

Answer:

(a) $\{6, 10\}$
(b) $\{1, 3, 5, 6, 7, 9, 10\}$
(c) $\emptyset$