# Families of sets indexed by another set

I am reading Automata and Computability by Dexter C. Kozen and I am in the first chapter entitled Strings and Sets.

If we have a set $$A=\{ab,b\}$$ do we get to assume that the null string $$\epsilon$$ is also a member of that set? If we have another set $$B=\{\epsilon\}$$, does $$A\cap B = \{\epsilon\}$$ or does $$A∩B=\emptyset$$?

On a related note, I am having trouble understanding the meaning of a little chunk in the textbook. What does "family of set indexed by another set $$I$$" mean? What does the set $$I$$ consist of here? I am very confused by the notation as well. Anything you can do to help me understand the excerpt below is appreciated.

Set concatenation distributes over union: \begin{align} A(B\cup C) &= AB \cup AC, \\ (A \cup B)C &= AC \cup BC. \end{align} In fact, concatenation distributes over the union of any family of sets. If $$\{B_i \mid i \in I \}$$ is any family of sets indexed by another set $$I$$, finite or infinite, then \begin{align} A \bigl( \bigcup_{i \in I} B_i \bigr) &= \bigcup_{i \in I} AB_i, \\ \bigl( \bigcup_{i \in I} B_i \bigr) A &= \bigcup_{i \in I} B_iA. \end{align} Here $$\bigcup_{i \in I} B_i$$ denotes the union of all the sets $$B_i$$ for $$i \in I$$. An element $$x$$ is in this union iff it is in one of the $$B_i$$.

• No, the empty string is a string that is not an element of $\{ab,b\}$ (if $a$ and $b$ are representing letters here).
– plop
Feb 2, 2021 at 23:15
• $I$ is just a set of indexes. For example, $I=\{1,2,3\}$ then $\bigcup_{i\in I}B_i$ is the same as $B_1\cup B_2\cup B_3$. But the set $I$ could be empty or it could be an infinite set, when you need to take a union of infinitely many sets.
– plop
Feb 2, 2021 at 23:16

The notation $$\{ x_1,x_2,x_3,\ldots,x_n \}$$ means "the set of elements $$x_1,x_2,x_3,\ldots,x_n$$". In particular, all elements in the set are listed. If an element isn't listed, it doesn't belong to the set. For example, $$\epsilon$$ is not an element of $$\{ab,b\}$$. There is absolutely nothing special about $$\epsilon$$. It is a bona fide word, and undergoes no special treatment. It is not an element of $$\{ab,b\}$$ just as $$abba$$ is not an element of $$\{ab,b\}$$.
As for your second question, here is a family of sets indexed by $$I = \{a,b,c\}$$: $$B_a = \{a,ba\}, B_b = \{b,ba\}, B_c = \{\epsilon,aa,bbb\}.$$ The union of this family is $$\bigcup_{i \in I} B_i = B_a \cup B_b \cup B_c = \{\epsilon, a, b, aa, ba, bbb \}.$$ The set $$I$$ is completely arbitrary. It could be the empty set, it could be a singleton, it could be a larger finite set, it could be an infinite set. For example, we can define a family of sets indexed by the natural numbers (including zero): for every $$n \in \mathbb{N}$$, $$S_n = \{n,-n\}$$. The union of this family consists of all integers: $$\bigcup_{n \in S_n} S_n = \mathbb{Z}$$,