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[tex]$D$[/tex] and [tex]$E$[/tex] are sets of real numbers defined as follows:
[tex]$
\begin{array}{l}
D=\{z \mid z \leq 4\} \\
E=\{z \mid z \ \textgreater \ 7\}
\end{array}
$[/tex]

Write [tex]$D \cap E$[/tex] and [tex]$D \cup E$[/tex] using interval notation. If the set is empty, write [tex]$\varnothing$[/tex].

[tex]$
D \cap E = \boxed{\varnothing}
$[/tex]
[tex]$
D \cup E = \boxed{(-\infty, 4] \cup (7, \infty)}
$[/tex]


Sagot :

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]