Let's analyze the problem from the image you provided. Here's the information from the problem:

• The set $C$ is defined as $C = \{ z \mid z \geq 2 \}$, meaning it includes all real numbers greater than or equal to 2.
• The set $D$ is defined as $D = \{ z \mid z \leq 5 \}$, meaning it includes all real numbers less than or equal to 5.

Union of $C$ and $D$ ( $C \cup D$ )

The union of two sets $C \cup D$ includes all elements that are in either set $C$ or set $D$, or in both.

• $C$ is all real numbers greater than or equal to 2, so it is the interval $[2, \infty)$.
• $D$ is all real numbers less than or equal to 5, so it is the interval $(-\infty, 5]$.

Thus, the union of these two sets will cover all real numbers because there is overlap between $C$ and $D$. So, $C \cup D = (-\infty, \infty)$, which includes all real numbers.

Intersection of $C$ and $D$ ( $C \cap D$ )

The intersection of two sets $C \cap D$ includes only the elements that are in both sets $C$ and $D$.

• $C$ is $[2, \infty)$, so it only includes numbers starting from 2.
• $D$ is $(-\infty, 5]$, so it only includes numbers up to 5.

The overlap between these two sets is the interval from 2 to 5. Therefore, $C \cap D = [2, 5]$.

Final Answer:

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

Let me know if you'd like further details or clarifications!

Here are 5 related questions for practice:

1. What is the union of $(-\infty, 0)$ and $[3, \infty)$?
2. How do you find the intersection of $(-2, 4]$ and $[0, \infty)$?
3. What does the intersection of $(-\infty, -3]$ and $(0, 5)$ look like?
4. If $A = [0, 3]$ and $B = [2, 6]$, what is $A \cap B$?
5. What is the union of $[0, 1]$ and $[2, 3]$?

Tip: When finding intersections, focus on the common overlapping range between the sets.