3. Let [tex]$U=\{1,2,3,4,5,6,7,8\}$[/tex], [tex]$A=\{1,3,4,5\}$[/tex], [tex][tex]$B=\{2,4,6,8\}$[/tex][/tex].

Write each of the given sets in roster form:

a. [tex]$A \cup B=$[/tex]

b. [tex]$A \cap B=$[/tex]

c. [tex][tex]$A^{\prime}=$[/tex][/tex]



Answer :

Let's solve each of the given sets step-by-step.

Given sets:
- The universal set [tex]\( U = \{1,2,3,4,5,6,7,8\} \)[/tex]
- Set [tex]\( A = \{1,3,4,5\} \)[/tex]
- Set [tex]\( B = \{2,4,6,8\} \)[/tex]

### a. [tex]\( A \cup B \)[/tex] (A union B)
The union of two sets, [tex]\( A \cup B \)[/tex], is the set of elements that are in either [tex]\( A \)[/tex], [tex]\( B \)[/tex], or both.

List the elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cup B = \{1, 2, 3, 4, 5, 6, 8\} \][/tex]

### b. [tex]\( A \cap B \)[/tex] (A intersection B)
The intersection of two sets, [tex]\( A \cap B \)[/tex], is the set of elements that are common to both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].

List the elements that are common in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cap B = \{4\} \][/tex]

### c. [tex]\( A^{\prime} \)[/tex] (Complement of A)
The complement of set [tex]\( A \)[/tex], denoted as [tex]\( A^{\prime} \)[/tex] or [tex]\( A^c \)[/tex], consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].

List the elements in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex]:
[tex]\[ A^{\prime} = \{2, 6, 7, 8\} \][/tex]

### Summary of Results
a. [tex]\( A \cup B = \{1, 2, 3, 4, 5, 6, 8\} \)[/tex]

b. [tex]\( A \cap B = \{4\} \)[/tex]

c. [tex]\( A^{\prime} = \{2, 6, 7, 8\} \)[/tex]