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.
 A: First you want a notation for repeating a string n times.
Let s be a string and define by induction, s^1 = s and s^(n+1) = ss^n.
Your set appears to be { x, y, yy, y^(4n) : n in N }.  
A: 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.
