Let's start by defining the required sets.
1. The universal set [tex]\( U \)[/tex]:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \][/tex]
2. Define set [tex]\( A \)[/tex] as:
[tex]\[ A = \{x \in U \mid x < 5 \} \][/tex]
This set includes all elements of [tex]\( U \)[/tex] that are less than 5:
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
3. Define set [tex]\( B \)[/tex] as:
[tex]\[ B = \{x \in U \mid x > 3 \} \][/tex]
This set includes all elements of [tex]\( U \)[/tex] that are greater than 3:
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]
Next, we need to find the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cap B \)[/tex]. The intersection of two sets is the set of elements that are common to both sets.
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]
The common element between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[ A \cap B = \{4\} \][/tex]
Now, we need to find the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cup B \)[/tex]. The union of two sets is the set of all elements that are in either set.
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]
Combining all elements from both sets without repeating any element, we get:
[tex]\[ A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \][/tex]
Therefore, the final answers are:
[tex]\[
\begin{array}{l}
A \cap B = \{4\} \\
A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
\end{array}
\][/tex]
