Given the image, we have the sets $F$ and $H$ defined as:

$F = \{ z \mid z \leq 4 \}$ $H = \{ z \mid z < 5 \}$

We are required to find the intersection ($F \cap H$) and the union ($F \cup H$) of these sets using interval notation.

1. Intersection ($F \cap H$):

• Set $F$ can be expressed in interval notation as $(-\infty, 4]$.
• Set $H$ can be expressed in interval notation as $(-\infty, 5)$.

The intersection of these intervals is the overlap:

$F \cap H = (-\infty, 4]$

2. Union ($F \cup H$):

• The union of these intervals combines all the values from both sets without duplication.

Since $(-\infty, 5)$ already covers $(-\infty, 4]$, the union is:

$F \cup H = (-\infty, 5)$

Final Answers:

• $F \cap H = (-\infty, 4]$
• $F \cup H = (-\infty, 5)$

Would you like further explanation or have any questions?

Related Questions:

1. How do you determine the intersection of two different types of intervals?
2. What is the difference between open and closed intervals?
3. How does interval notation represent inequalities?
4. How can you visualize interval notation on a number line?
5. What happens if the intervals do not overlap when finding an intersection?

Tip: Remember, an interval that includes its boundary is closed, denoted by square brackets $[ ]$, while an interval that does not include its boundary is open, denoted by parentheses $( )$.