Let's carefully examine the given sets and perform the required set operations to complete the activity as follows:
Given sets:
- Universal set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13\} \)[/tex]
- Set [tex]\( A = \{1, 2, 3, 5, 7\} \)[/tex]
- Set [tex]\( B = \{1, 5, 8, 9, 10\} \)[/tex]
Firstly, we need to find the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The union of two sets [tex]\( A \cup B \)[/tex] includes all elements that are in either set [tex]\( A \)[/tex], set [tex]\( B \)[/tex], or both. The union operation results in the set of all distinct elements from both sets.
So, the union [tex]\( A \cup B \)[/tex] is:
[tex]\[ A \cup B = \{1, 2, 3, 5, 7, 8, 9, 10\} \][/tex]
Next, we need to find the cardinality (number of elements) of [tex]\( A \cup B \)[/tex], denoted as [tex]\( n(A \cup B) \)[/tex]:
[tex]\[ n(A \cup B) = 8 \][/tex]
Now, we will determine the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The intersection [tex]\( A \cap B \)[/tex] includes only the elements that are in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Thus, the intersection [tex]\( A \cap B \)[/tex] is:
[tex]\[ A \cap B = \{1, 5\} \][/tex]
Finally, we need to find the cardinality (number of elements) of [tex]\( A \cap B \)[/tex], denoted as [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ n(A \cap B) = 2 \][/tex]
To summarize, the completed activity would be:
[tex]\[
\begin{array}{l}
U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13\} \\
A = \{1, 2, 3, 5, 7\} \\
B = \{1, 5, 8, 9, 10\} \\
A \cup B = \{1, 2, 3, 5, 7, 8, 9, 10\} \\
n(A \cup B) = 8 \\
A \cap B = \{1, 5\} \\
n(A \cap B) = 2 \\
\end{array}
\][/tex]
