The problem asks you to use set-builder notation to represent the set $H$, defined as the set of real numbers greater than six.

In set-builder notation, we would write this as: $H = \{ x \mid x > 6, \, x \in \mathbb{R} \}$

This reads as "H is the set of all $x$ such that $x$ is greater than 6, and $x$ is a real number."

Would you like further details on set-builder notation or have any questions?

Here are some related questions that could help deepen understanding:

1. How would you represent a set of real numbers less than a certain value in set-builder notation?
2. What does the symbol $\mathbb{R}$ represent in mathematics?
3. How would the notation change if we included 6 in the set (i.e., $x \geq 6$)?
4. How could you represent an interval for all real numbers between two values, like 3 and 7?
5. How would the notation change if we were working with integers instead of real numbers?

Tip: In set-builder notation, the vertical bar $|$ or colon $:$ means "such that" and is used to define the property that elements of the set must satisfy.