1

Section 1.9 Binary Strings of the textbook Introduction to Lattices and Order, second edition, by Davey and Priestley, says the following:

Let $\Sigma^\ast$ be the set of all finite binary strings, that is, all finite sequences of zeros and ones; the empty string is included. Adding the infinite sequences, we get the set of all finite or infinite sequences, which we denote $\Sigma^{\ast \ast}$.

I'm confident that, as alluded to, the set of all finite binary strings is a set of infinite sequences (in terms of the amount of finite sequences in the set). But does adding all of these sequences, which, as stated, would be an infinite amount of sequences, actually get us the set of all infinite sequences, as it claims? It intuitively seems like it would be true, but intuition is often wrong when reasoning about infinity in such ways, so I'm not totally sure.

The Pointer
  • 217
  • 1
  • 10
  • 1
    The set of all finite binary strings is a set of finite sequences. Adding the set of all infinite sequences results in $\Sigma^{*}$, by definition*. – Yuval Filmus Dec 30 '23 at 18:55
  • @YuvalFilmus But there are an infinite number of finite binary strings, so the set of all finite binary strings has an infinite number of elements, no? Isn't that what the author is referring to when they say "Adding the infinite sequences" after "Let $\Sigma^\ast$ be the set of all finite binary strings, that is, all finite sequences of zeros and ones; the empty string is included."? – The Pointer Dec 31 '23 at 01:44
  • 1
    When the author refers to infinite sequences, they refer to sequences which are individually infinite, like the decimal expansion of a real number. – Yuval Filmus Dec 31 '23 at 05:55
  • @YuvalFilmus Ohhh, I see what it means: By "Adding the infinite sequences", it means the set of all infinite sequences in addition to the set of all finite binary strings (that it just mentioned). I misinterpreted it. Thanks for the clarification. – The Pointer Dec 31 '23 at 10:50

1 Answers1

0

You can prove this to yourself formally as follows: Say that $F$ is the set of finite binary strings, and $I$ is the set of infinite binary strings. Both of these sets are infinite. Now, the question is whether $I\cup F$ is the set of all binary strings. This is equivalent to saying that for any binary string $b$, $b\in I\cup F$. Take some such binary string $b$. $b$ must be either infinite or finite. If $b$ is infinite, $b\in I$, so $b\in I\cup F$. If $b$ is finite, $b\in F$, so $b\in I\cup F$. Therefore, for every $b$, we know that $b\in I\cup F$. Therefore, we know that $I\cup F$ (or $\Sigma^{**}$ as the book calls it) is the set of all binary strings.

nosyarg
  • 232
  • 1
  • 8