From the definitions given in the image:

• $E = \{ x \mid x > 1 \}$: This is the set of real numbers greater than 1, which can be written as $(1, \infty)$ in interval notation.
• $F = \{ x \mid x \leq 8 \}$: This is the set of real numbers less than or equal to 8, which can be written as $(-\infty, 8]$ in interval notation.

Solution:

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

   • The intersection contains elements that satisfy both $x > 1$ and $x \leq 8$.
   • Combining these, $E \cap F = (1, 8]$.

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

   • The union contains all elements in either $x > 1$ or $x \leq 8$.
   • This encompasses all real numbers since $x > 1$ and $x \leq 8$ overlap and cover the entire real line.
   • Thus, $E \cup F = (-\infty, \infty)$.

Final Answer in Interval Notation:

• $E \cap F = (1, 8]$
• $E \cup F = (-\infty, \infty)$

Let me know if you need further clarification or explanation!

Questions for deeper understanding:

1. What is the geometric interpretation of interval notation on the number line?
2. How do intersections differ from unions in terms of logic?
3. Could the interval bounds change if $E$ and $F$ included equality (e.g., $x \geq 1$ or $x < 8$)?
4. How would you solve this if the sets $E$ and $F$ had no overlap?
5. How can this concept be extended to three or more sets?

Tip: Always verify boundaries when determining intersections or unions to ensure accuracy in interval notation.