The problem asks to write the union $E \cup F$ and the intersection $E \cap F$ using interval notation, where the sets $E$ and $F$ are defined as follows:

$E = \{ x \mid x < 1 \}$ $F = \{ x \mid x \geq 6 \}$

Step 1: Union ($E \cup F$)

The union of two sets includes all elements that are in either set.

• For $E$, all real numbers less than 1 are included, i.e., $(-\infty, 1)$.
• For $F$, all real numbers greater than or equal to 6 are included, i.e., $[6, \infty)$.

Thus, the union of $E$ and $F$ is: $E \cup F = (-\infty, 1) \cup [6, \infty)$

Step 2: Intersection ($E \cap F$)

The intersection of two sets includes only the elements that are in both sets.

• For $E$, the elements are less than 1.
• For $F$, the elements are greater than or equal to 6.

Since there is no overlap between these two sets, their intersection is empty: $E \cap F = \emptyset$

So, the answers are:

• $E \cup F = (-\infty, 1) \cup [6, \infty)$
• $E \cap F = \emptyset$

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

Here are 5 related questions you can explore:

1. What is the union of $(-\infty, 0)$ and $[2, \infty)$?
2. How do you find the intersection of two overlapping intervals, like $(-\infty, 3)$ and $[1, 5)$?
3. Can two disjoint sets ever have a non-empty intersection?
4. How do you represent an empty set in set notation?
5. How do interval notations differ for open and closed sets?

Tip: When working with unions and intersections, always visualize the number line to clearly see where the intervals overlap or combine.