To determine [tex]\(D \cup E\)[/tex] and [tex]\(D \cap E\)[/tex] using interval notation, let's first define the sets [tex]\(D\)[/tex] and [tex]\(E\)[/tex] more explicitly:
1. Set [tex]\(D\)[/tex]:
[tex]\[
D = \{w \mid w > 1\}
\][/tex]
This set includes all real numbers greater than 1. In interval notation, it is written as:
[tex]\[
D = (1, \infty)
\][/tex]
2. Set [tex]\(E\)[/tex]:
[tex]\[
E = \{w \mid w \leq 7\}
\][/tex]
This set includes all real numbers less than or equal to 7. In interval notation, it is written as:
[tex]\[
E = (-\infty, 7]
\][/tex]
Next, let's find the union and intersection of these sets.
### Union [tex]\(D \cup E\)[/tex]
The union of two sets [tex]\(D \cup E\)[/tex] includes all elements that are in [tex]\(D\)[/tex] or [tex]\(E\)[/tex] or both. In this case:
- [tex]\(D\)[/tex] contains all numbers greater than 1.
- [tex]\(E\)[/tex] contains all numbers less than or equal to 7.
Combining these two intervals, we cover all real numbers except the single point [tex]\(1\)[/tex]:
[tex]\[
D \cup E = (-\infty, \infty)
\][/tex]
Since there are no gaps between the two intervals and they collectively cover all real numbers, the union is the set of all real numbers.
### Intersection [tex]\(D \cap E\)[/tex]
The intersection of two sets [tex]\(D \cap E\)[/tex] includes all elements that are in both [tex]\(D\)[/tex] and [tex]\(E\)[/tex]. In this case:
- For a number to be in [tex]\(D\)[/tex], it must be greater than 1.
- For a number to be in [tex]\(E\)[/tex], it must be less than or equal to 7.
Thus, the overlap between these two sets consists of all numbers greater than 1 and less than or equal to 7:
[tex]\[
D \cap E = (1, 7]
\][/tex]
### Final Results
[tex]\[
D \cup E = (-\infty, \infty)
\][/tex]
[tex]\[
D \cap E = (1, 7]
\][/tex]
