Let's analyze the problem step by step.

You are given two sets of real numbers, defined as follows:

• $C = \{ z \mid z > 2 \}$, which means all real numbers greater than 2.
• $D = \{ z \mid z \leq 5 \}$, which means all real numbers less than or equal to 5.

You are asked to find:

1. $C \cup D$ (the union of sets $C$ and $D$)
2. $C \cap D$ (the intersection of sets $C$ and $D$).

1. Union: $C \cup D$

The union of two sets is the set of elements that belong to either set. Set $C$ consists of numbers greater than 2, and set $D$ consists of numbers less than or equal to 5. So:

• $C = (2, \infty)$
• $D = (-\infty, 5]$

Thus, their union covers all numbers from $-\infty$ to $\infty$, meaning every real number is included. Hence:

$C \cup D = (-\infty, \infty)$

2. Intersection: $C \cap D$

The intersection of two sets is the set of elements that belong to both sets. For this:

• Numbers in $C$ are greater than 2.
• Numbers in $D$ are less than or equal to 5.

Therefore, the intersection consists of numbers that are greater than 2 and less than or equal to 5. So:

$C \cap D = (2, 5]$

Correct Answers

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

Looking at your screenshot, it seems like the issue with your answer is related to the incorrect use of brackets. The correct interval notation for the intersection should be $(2, 5]$, using a parenthesis on the left because 2 is not included in $C$, and a square bracket on the right because 5 is included in $D$.

Would you like more details or have any other questions? Here are some related questions to further your understanding:

1. How is the union of sets determined using interval notation?
2. What is the difference between a union and an intersection of sets?
3. How do we decide when to use square brackets versus parentheses in interval notation?
4. Can an intersection of two sets ever result in an empty set?
5. What happens if the intervals do not overlap at all?

Tip: When writing interval notation, always pay attention to whether endpoints are included or excluded from the interval!