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Chapter 13: Problem 8

Suppose that \(A\) is a subset of \(V^{*},\) where \(V\) is an alphabet. Prove or disprove each of these statements. $$\begin{array}{ll}{\text { a) } A \subseteq A^{2}} & {\text { b) if } A=A^{2}, \text { then } \lambda \in A} \\ {\text { c) } A\\{\lambda\\}=A} & {\text { d) }\left(A^{*}\right)^{*}=A^{*}} \\ {\text { e) } A^{*} A=A^{*}} & {\text { f }\left|A^{n}\right|=|A|^{n}}\end{array}$$

Short Answer

Expert verified
a) Disprove, b) Prove, c) Disprove, d) Prove, e) Prove, f) Prove.

Step by step solution

01

Analyze Statement (a)

Statement (a) claims that every element of set A is also in the set A虏. Recall that A虏 is formed by concatenating elements of A with each other. We need to consider if every element in A can be written as a concatenation of two elements from A. This is generally not true unless A includes some form of the empty string or its elements can inherently be concatenated to form themselves.
02

Evaluate Statement (b)

Statement (b) claims that if A equals A虏, then the empty string 位 must be in A. If A equals A虏, this means every element in A can be represented as a concatenation of two elements from A. The only way this can be true for all elements in A is if 位 is in A, allowing any element to concatenate with 位 to return itself.
03

Analyze Statement (c)

Statement (c) states that removing 位 from A leaves the set unchanged, which is not generally true. The set A could contain 位, and removing it would change the set.
04

Evaluate Statement (d)

Statement (d) concerns the closure properties of the Kleene star. Recall that A* is the set of all strings formed by concatenating zero or more elements from A. By definition of the Kleene star operator, applying it twice does not change the result. Hence (A*)* = A*.
05

Analyze Statement (e)

Statement (e) suggests that concatenating A* with A results in A*. By properties of the Kleene star, strings in A* already include all possible concatenations of elements in A, including those formed by concatenating an additional element from A.
06

Evaluate Statement (f)

Statement (f) concerns the cardinality of A to the power n. If A has |A| elements, then the set of all n-length strings that can be formed from A, denoted by An, has |A|^n elements.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set theory is a fundamental concept in mathematics that deals with the collection of objects, known as sets. Sets are defined as unordered collections of distinct elements. In the exercise, sets are used to represent collections of strings over an alphabet.

Here are some key points to understand:
  • A set is usually denoted with capital letters, such as A, B, C.
  • Elements of a set can be anything: numbers, characters, strings, etc.
  • A subset, denoted by \(A \subseteq B\), means every element of set A is also in set B.
  • The empty set, denoted by \(\lambda\), contains no elements.
  • The union of sets \(A \cup B\) is the set containing all elements of A and B.
  • The intersection of sets \(A \cap B\) is the set containing elements common to both A and B.
In this exercise, we use the concept of subsets to analyze different set properties and operations.
Concatenation
Concatenation is a basic operation in string manipulation that involves joining two strings end-to-end. It is a fundamental concept in automata theory and formal languages.

In the context of set theory and languages, concatenation can be applied to sets of strings to form new strings:
  • If A and B are sets of strings, their concatenation is represented by \(A \cdot B\).
  • Each string in the resulting set is formed by taking a string from A and following it with a string from B.
For example, if \(A = {a, b}\) and \(B = {c, d}\), then \(A \cdot B = {ac, ad, bc, bd}\).

In the exercise, we see *concatenation* in statements like \(A^2\) and \(A*\). Here, \(A^2\) represents the concatenation of set A with itself, which forms strings by concatenating any two elements from A.
Kleene Star
The Kleene star operator is a key concept in formal language theory, represented by the symbol \( * \). It denotes the set of all strings that can be formed by concatenating zero or more elements from a set. If A is a set of strings, then \(A*\) includes:
  • The empty string \( \lambda \).
  • All individual elements of A.
  • All possible concatenations of these elements, repeated any number of times.
For instance, if \( A = {a, b} \), then \( A* = { \lambda, a, b, aa, ab, ba, bb, aaa, aab, ...} \).

In the exercise, it's mentioned that \( (A*)* = A* \), which is true by definition: applying the Kleene star operator twice does not change the set, as the result still includes all possible concatenations defined by * once.
Cardinality
Cardinality refers to the number of elements in a set. It is an important measure in set theory often denoted by \( |A| \) for a set A. Understanding the cardinality helps us analyze the size and properties of sets.

Key points about cardinality include:
  • If a set A has 3 elements, then \( |A| = 3 \).
  • For the power set (the set of all subsets) of a set A with n elements, the cardinality is \( 2^n \).
In the context of the exercise, Statement (f) claims that the cardinality of \( A^n \) is \( |A|^n \). This is because forming n-length strings from set A involves choosing each of the n positions independently from the |A| elements, resulting in \( |A|^n \) different combinations.

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Most popular questions from this chapter

Express each of these sets using a regular expression. a) the set consisting of the strings \(0,11,\) and 010 b) the set of strings of three 0 s followed by two or more 0 \(\mathrm{s}\) c) the set of strings of odd length d) the set of strings that contain exactly one 1 e) the set of strings ending in 1 and not containing 000

Construct a Turing machine that recognizes the set of all bit strings that end with a 0.

Construct a Turing machine with tape symbols 0, 1, and B that, given a bit string as input, replaces the first two consecutive 1s on the tape with 0s and does not change any of the other symbols on the tape.

Find a phrase-structure grammar for each of these languages. a) the set of all bit strings containing an even number of 0 s and no 1 \(\mathrm{s}\) b) the set of all bit strings made up of a 1 followed by an odd number of 0s c) the set of all bit strings containing an even number of 0s and an even number of 1s d) the set of all strings containing 10 or more 0s and no 1s e) the set of all strings containing more 0s than 1s f ) the set of all strings containing an equal number of 0s and 1s g) the set of all strings containing an unequal number of 0s and 1s

Describe in words the strings in each of these regular sets. \(\begin{array}{ll}{\text { a) } 1^{*} 0} & {\text { b) } 1^{*} 00^{*}} \\\ {\text { c) } 111 \cup 001} & {\text { d) }(1 \cup 00)^{*}} \\ {\text { e) }\left(00^{*} 1\right)^{*}} & {\text { f) }(0 \cup 1)(0 \cup 1)^{*} 00}\end{array}\)
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