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.

  • $\begingroup$ Your problem is not how to use set notation for sets with repeating characters -- your problem is how to notate that a string has repeating characters. $\endgroup$ – user14972 Sep 24 '17 at 22:32
  • $\begingroup$ I was thinking try something like let $W_n(str)$ mean the value of str repeated n times then you could use that notation to specify things in the set builder notation. $\endgroup$ – user451844 Sep 24 '17 at 22:33
  • $\begingroup$ $\{w \mid w=x \lor \exists n \in \Bbb N [w=y^{2^n}]\}$? $\endgroup$ – Kenny Lau Sep 24 '17 at 22:33
  • 1
    $\begingroup$ $\{x\} \cup \{y^{2^n}: n\in\mathbb{N}\}$ could also work depending on context and level of formality $\endgroup$ – Brevan Ellefsen Sep 24 '17 at 22:36
  • $\begingroup$ Do you intend for $yy\dots y$ to be a string, or some multiplicative operation applied repeatedly? $\endgroup$ – Brevan Ellefsen Sep 24 '17 at 22:37

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 }.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.