The problem provides two sets of real numbers:

• $D = \{ z \mid z > 2 \}$, which is the set of numbers greater than 2.
• $E = \{ z \mid z \leq 5 \}$, which is the set of numbers less than or equal to 5.

We are asked to find:

1. $D \cup E$ (the union of sets $D$ and $E$) using interval notation.
2. $D \cap E$ (the intersection of sets $D$ and $E$) using interval notation.

Step 1: Find $D \cup E$ (Union)

• $D$ represents all real numbers greater than 2, so in interval notation: $D = (2, \infty)$.
• $E$ represents all real numbers less than or equal to 5, so in interval notation: $E = (-\infty, 5]$.
• The union $D \cup E$ consists of all numbers that are in $D$ or $E$, which covers from negative infinity up to infinity. Therefore, $D \cup E = (-\infty, \infty)$.

Step 2: Find $D \cap E$ (Intersection)

• The intersection $D \cap E$ consists of numbers that are in both $D$ and $E$.
• $D = (2, \infty)$ and $E = (-\infty, 5]$.
• The common part of these two sets is the interval where $z > 2$ and $z \leq 5$, which is $(2, 5]$. Therefore, $D \cap E = (2, 5]$.

Final Answer:

• $D \cup E = (-\infty, \infty)$
• $D \cap E = (2, 5]$

Would you like further clarification or details? Here are some related questions:

1. How do you represent other types of sets, like $D$ and $E$, in interval notation?
2. What is the difference between union and intersection of sets?
3. Can the intersection of sets ever be empty? If so, under what conditions?
4. How do you express inequalities involving strict and non-strict conditions in set notation?
5. Why does $D \cup E$ span all real numbers in this case?

Tip: In set notation, $\cup$ represents a union, including elements from either set, while $\cap$ represents an intersection, including only elements common to both sets.