Question

let ( u ) be the universal set, where:
( u = {1,2,3,4,5,6,7,8,9,10,11,12,13} )
let sets ( a ), ( b ), and ( c ) be subsets of ( u ), where:
( a = {3,8,9,10,11} )
( b = {4,8,10,11,13} )
( c = {5,9,12,13} )
find the following:

list the elements in the set ( a cup b ):
( a cup b = { )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne

list the elements in the set ( a cap b ):
( a cap b = { )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne

enter the elements in the set ( c cup a ):
( c cup a = { )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne

enter the elements in the set ( (a cap b) cap c ):
( (a cap b) cap c = { )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne

Explanation:

Step1: Find \( A' \) (complement of \( A \))

The universal set \( U = \{1,2,3,4,5,6,7,8,9,10,11,12,13\} \), set \( A = \{3,8,9,10,11\} \). The complement \( A' \) contains elements in \( U \) but not in \( A \). So \( A' = \{1,2,4,5,6,7,12,13\} \).

Step2: Find \( A' \cup B \)

Set \( B = \{4,8,10,11,13\} \). The union of \( A' \) and \( B \) contains all elements in \( A' \) or \( B \). Combining \( A' = \{1,2,4,5,6,7,12,13\} \) and \( B = \{4,8,10,11,13\} \), we get \( A' \cup B = \{1,2,4,5,6,7,8,10,11,12,13\} \).

Step3: Find \( A \cap B \)

Set \( A = \{3,8,9,10,11\} \), set \( B = \{4,8,10,11,13\} \). The intersection contains elements common to both \( A \) and \( B \). So \( A \cap B = \{8,10,11\} \).

Step4: Find \( C' \) (complement of \( C \))

Set \( C = \{5,9,12,13\} \), universal set \( U \). So \( C' = \{1,2,3,4,6,7,8,10,11\} \).

Step5: Find \( C' \cup A \)

Set \( A = \{3,8,9,10,11\} \). Union of \( C' \) and \( A \): combining \( C' = \{1,2,3,4,6,7,8,10,11\} \) and \( A = \{3,8,9,10,11\} \), we get \( C' \cup A = \{1,2,3,4,6,7,8,9,10,11\} \).

Step6: Find \( (A \cap B)' \)

First, \( A \cap B = \{8,10,11\} \) (from Step3). The complement of this set in \( U \) is \( (A \cap B)' = \{1,2,3,4,5,6,7,9,12,13\} \).

Step7: Find \( (A \cap B) \cap C' \)

First, \( A \cap B = \{8,10,11\} \) (Step3), \( C' = \{1,2,3,4,6,7,8,10,11\} \) (Step4). The intersection of these two sets is \( \{8,10,11\} \).

Answer:

s:

• \( A' \cup B \): \( 1,2,4,5,6,7,8,10,11,12,13 \)
• \( A \cap B \): \( 8,10,11 \)
• \( C' \cup A \): \( 1,2,3,4,6,7,8,9,10,11 \)
• \( (A \cap B)' \): \( 1,2,3,4,5,6,7,9,12,13 \)
• \( (A \cap B) \cap C' \): \( 8,10,11 \)