Question

question 1
let ( u = {1,2,3,dots,8,9,10} ) be the universal set.
let sets ( a ) and ( b ) be subsets of ( u ), where:
set ( a = {2,5,6,8,10} )
set ( b = {2,4,5,8} )
list the elements in the set ( a ):
( a = { ) (}
list the elements in the set ( b ):
( b = { ) (}
list the elements in the set ( a cup b ):
( a cup b = { ) (}
list the elements in the set ( a cap b ):
( a cap b = { ) (}

Explanation:

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

The complement of a set \( A \) (denoted \( A' \)) consists of all elements in the universal set \( U \) that are not in \( A \). The universal set \( U=\{1,2,3,4,5,6,7,8,9,10\} \) and set \( A = \{2,5,6,8,10\} \). So we remove the elements of \( A \) from \( U \).
Elements in \( U \) not in \( A \): \( 1,3,4,7,9 \)
Thus, \( A'=\{1,3,4,7,9\} \)

Step2: Find \( B' \) (complement of \( B \))

The complement of set \( B \) (denoted \( B' \)) consists of all elements in \( U \) not in \( B \). Set \( B = \{2,4,5,8\} \). Remove elements of \( B \) from \( U \).
Elements in \( U \) not in \( B \): \( 1,3,6,7,9,10 \)
Thus, \( B'=\{1,3,6,7,9,10\} \)

Step3: Find \( A \cup B \) (union of \( A \) and \( B \))

The union of two sets \( A \) and \( B \) (denoted \( A \cup B \)) consists of all elements that are in \( A \) or in \( B \) (or in both). Combine the elements of \( A \) and \( B \), removing duplicates.
\( A = \{2,5,6,8,10\} \), \( B = \{2,4,5,8\} \)
Combined elements: \( 2,4,5,6,8,10 \)
Thus, \( A \cup B=\{2,4,5,6,8,10\} \)

Step4: Find \( A \cap B \) (intersection of \( A \) and \( B \))

The intersection of two sets \( A \) and \( B \) (denoted \( A \cap B \)) consists of all elements that are in both \( A \) and \( B \). Find common elements in \( A \) and \( B \).
Common elements in \( A = \{2,5,6,8,10\} \) and \( B = \{2,4,5,8\} \): \( 2,5,8 \)
Thus, \( A \cap B=\{2,5,8\} \)

Answer:

\( A' = \{1, 3, 4, 7, 9\} \)
\( B' = \{1, 3, 6, 7, 9, 10\} \)
\( A \cup B = \{2, 4, 5, 6, 8, 10\} \)
\( A \cap B = \{2, 5, 8\} \)