To find the sets [tex]\( A \cup B \)[/tex] and [tex]\( A \cap B \)[/tex], we need to understand the operations of union and intersection between the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### Step-by-Step Solution
1. Understanding Union ([tex]\( A \cup B \)[/tex]):
The union of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of all elements that are in [tex]\( A \)[/tex], in [tex]\( B \)[/tex], or in both sets. In other words, [tex]\( A \cup B \)[/tex] includes every unique element from both sets.
2. Understanding Intersection ([tex]\( A \cap B \)[/tex]):
The intersection of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of all elements that are both in [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### Given Sets:
- [tex]\( A = \{2, 4, 5, 10, 13, 17, 19\} \)[/tex]
- [tex]\( B = \{1, 4, 6, 8, 9, 10, 13, 14, 15, 16, 17\} \)[/tex]
3. Finding [tex]\( A \cup B \)[/tex] (Union):
We list all unique elements that are either in [tex]\( A \)[/tex], [tex]\( B \)[/tex], or both.
[tex]\[
A \cup B = \{1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 16, 17, 19\}
\][/tex]
4. Finding [tex]\( A \cap B \)[/tex] (Intersection):
We list the elements that are common in both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\[
A \cap B = \{4, 10, 13, 17\}
\][/tex]
### Final Answer
- Elements in the set [tex]\( A \cup B \)[/tex]:
[tex]\[
A \cup B = \{1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 16, 17, 19\}
\][/tex]
- Elements in the set [tex]\( A \cap B \)[/tex]:
[tex]\[
A \cap B = \{4, 10, 13, 17\}
\][/tex]
