To find [tex]\((A \cup B)^{\prime}\)[/tex], follow these steps:
1. Identify the Universal Set [tex]\( U \)[/tex] and Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Universal Set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- Set [tex]\( A = \{2, 4, 6, 8, 10\} \)[/tex]
- Set [tex]\( B = \{1, 2, 3, 4, 5, 6\} \)[/tex]
2. 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 in both.
- [tex]\( A \cup B = \{2, 4, 6, 8, 10\} \cup \{1, 2, 3, 4, 5, 6\} \)[/tex]
- Combine all unique elements from both sets: [tex]\( A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\} \)[/tex]
3. Find the Complement of [tex]\( A \cup B \)[/tex] in the Universal Set [tex]\( U \)[/tex]:
- The complement of a set [tex]\( (A \cup B)^{\prime} \)[/tex] consists of all the elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \cup B \)[/tex].
- Identify the elements in [tex]\( U \)[/tex] that are not in [tex]\( A \cup B \)[/tex]:
- Universal Set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- [tex]\( A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\} \)[/tex]
- Elements in [tex]\( U \)[/tex] but not in [tex]\( A \cup B \)[/tex]: [tex]\( \{7, 9\} \)[/tex]
4. Conclusion:
Therefore, the complement of [tex]\( A \cup B \)[/tex], denoted as [tex]\( (A \cup B)^{\prime} \)[/tex], is [tex]\( \{7, 9\} \)[/tex].
