To determine whether pairs of sets are disjoint, we need to check if they have no elements in common. Let's analyze each pair of sets given:
1. [tex]\( A = \{4, 5, 6, 9, 10\} \)[/tex] and [tex]\( B = \{6, 11, 14, 17\} \)[/tex]
- Check for common elements:
- The element 6 is in both sets.
- Since they share the element 6, these sets are not disjoint.
2. [tex]\( A = \{ x \mid x \in \mathbb{Z} \text{ and } x \text{ is odd} \} \)[/tex] and [tex]\( B = \{ x \mid x \in \mathbb{R}, \ 8 \leq x \leq 9 \} \)[/tex]
- Set [tex]\( A \)[/tex] consists of all odd integers.
- Set [tex]\( B \)[/tex] consists of all real numbers in the interval [tex]\([8, 9]\)[/tex], which are not integers and hence cannot be odd integers.
- Thus, there are no common elements, and these sets are disjoint.
3. [tex]\( A = \{1, 3, 4, 6\} \)[/tex] and [tex]\( B = \{9, 11, 13, 14, 17\} \)[/tex]
- Check for common elements:
- There are no shared elements between the two sets.
- Since they have no elements in common, these sets are disjoint.
4. [tex]\( A = \{ x \mid x \in \mathbb{Z} \text{ and } x \text{ is even} \} \)[/tex] and [tex]\( B = \{ x \mid x \in \mathbb{R}, \ 6 < x < 8 \} \)[/tex]
- Set [tex]\( A \)[/tex] consists of all even integers.
- Set [tex]\( B \)[/tex] consists of all real numbers between 6 and 8, which are not integers and hence cannot be even integers.
- Thus, there are no common elements, and these sets are disjoint.
### Conclusion:
The pairs of sets that are disjoint are:
- [tex]\( \{1, 3, 4, 6\} \)[/tex] and [tex]\( \{9, 11, 13, 14, 17\} \)[/tex]
- [tex]\( \{ x \mid x \in \mathbb{Z} \text{ and } x \text{ is even} \} \)[/tex] and [tex]\( \{ x \mid x \in \mathbb{R}, \ 6 < x < 8 \} \)[/tex]
