Using set notation for sets with repeating characters Consider the following set:
{x, y, yy, yyyy, yyyyyyyy, yyyyyyyyyyyyyyyy, ...}
How would set builder notation be used to represent such a set? My understanding of the limitation is very limited as I only have experience using it for sets whose only elements are numbers, but never with character strings.
I also do not understand how to specify just one starting character that doesn't seem to be involved in any pattern.

It depends on what symbol juxtaposition means. Assuming it means multiplication, use this:
$$\{x\}\cup\bigcup_{n=0}^\infty \left\{ y^{\left( 2^n\right)} \right\}\tag{1}$$
The part on the right means
$$\left\{ y^{\left( 2^0\right)} \right\} \cup \left\{ y^{\left( 2^1\right)} \right\} \cup \left\{ y^{\left( 2^2\right)} \right\} \cup \left\{ y^{\left( 2^3\right) }\right\} \cup \cdots = \left\{ y, y^2, y^4, y^8, \cdots\right\}$$
The “$\{x\}\cup$” part on the left is because $x$ doesn’t seem to fit the definition of the pattern.
I used braces so much, because you can only unite sets. If your elements are themselves sets of elements, then uniting them without braces gives you a set of elements, while uniting them with braces gives a set of sets of elements.
If juxtaposition doesn’t mean multiplication, you will just need to define your own notation and adjust $(1)$ accordingly.
