To determine the union and intersection of the sets [tex]\( D \)[/tex] and [tex]\( E \)[/tex], let's first understand the definitions of these sets:
- The set [tex]\( D \)[/tex] consists of all real numbers [tex]\( z \)[/tex] such that [tex]\( z \leq 4 \)[/tex]. In interval notation, this can be written as:
[tex]\[
D = (-\infty, 4]
\][/tex]
- The set [tex]\( E \)[/tex] consists of all real numbers [tex]\( z \)[/tex] such that [tex]\( z > 7 \)[/tex]. In interval notation, this can be written as:
[tex]\[
E = (7, \infty)
\][/tex]
### Intersection [tex]\( D \cap E \)[/tex]
The intersection of two sets [tex]\( D \)[/tex] and [tex]\( E \)[/tex], denoted [tex]\( D \cap E \)[/tex], is the set of elements that are in both [tex]\( D \)[/tex] and [tex]\( E \)[/tex].
To find [tex]\( D \cap E \)[/tex], we need to identify the elements that satisfy both conditions:
- [tex]\( z \leq 4 \)[/tex]
- [tex]\( z > 7 \)[/tex]
There are no real numbers that simultaneously satisfy [tex]\( z \leq 4 \)[/tex] and [tex]\( z > 7 \)[/tex]. Therefore, the intersection is the empty set. In interval notation, this is denoted by:
[tex]\[
D \cap E = \varnothing
\][/tex]
### Union [tex]\( D \cup E \)[/tex]
The union of two sets [tex]\( D \)[/tex] and [tex]\( E \)[/tex], denoted [tex]\( D \cup E \)[/tex], is the set of elements that are in either [tex]\( D \)[/tex] or [tex]\( E \)[/tex] or both.
To find [tex]\( D \cup E \)[/tex], we combine all elements from both sets:
- [tex]\( D = (-\infty, 4] \)[/tex]
- [tex]\( E = (7, \infty) \)[/tex]
The elements in the union will satisfy either [tex]\( z \leq 4 \)[/tex] or [tex]\( z > 7 \)[/tex]. Therefore, in interval notation:
[tex]\[
D \cup E = (-\infty, 4] \cup (7, \infty)
\][/tex]
So, the answers are:
[tex]\[
D \cap E = \varnothing
\][/tex]
[tex]\[
D \cup E = (-\infty, 4] \cup (7, \infty)
\][/tex]
