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Chapter 0: Q9E (page 27)

Write a formal description of the following graph.

Short Answer

Expert verified

The formal description of the graph is:

{{1,2,3,4,5,6},{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}}

Step by step solution

01

Describe the parts of the formal description of the graph

A graph G can be defined as the collection of finite vertices and edges.

The set of finite vertices V are {1,2,3,4,5,6}.

The set of finite edges are {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}.

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Most popular questions from this chapter

Write formal descriptions of the following sets.

  1. The set containing the numbers1,10, and100
  2. The set containing all integers that are greater than5
  3. The set containing all natural numbers that are less than5
  4. The set containing the string aba
  5. The set containing the empty string
  6. The set containing nothing at all

In certain programming languages, comments appear between delimiters such as #and #/. Let Cbe the language of all valid delimited comment strings. A member of Cmust begin with #/and end with #/but have no intervening #/. For simplicity, assume that the alphabet for Cis 鈭�={a,b,/,#}.

a. Give aDFA that recognizes C.

b. Give a regular expression that generates C.

A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form

未 : Q 脳 螕鈭掆啋Q 脳 螕 脳 {R, RESET}.

If 未(q, a) = (r, b, RESET), when the machine is in state q reading an a, the machine鈥檚 head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.

Myhill鈥揘erode theorem. Refer to Problem 1.51 . Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pair wise distinguishable by L . The index of L may be finite or infinite.

a. Show that if L is recognized by a DFA with k states, L has index at most k.

b. Show that if the index of L is a finite number K , it is recognized by a DFA with k states.

c. Conclude that L is regular iff it has finite index. Moreover, its index is the size of the smallest DFA recognizing it.

Let X be the set {1,2,3,4,5}and Y be the set {6,7,8,9,10}.The unary function f:X鈫�Yand the binary function g:X脳Y鈫�Yare described in the following tables.

g12345f(n)67676 g123456789101010101010789106789106789106789106

a. What is the value of f(2)?
b.What are the range and domain of f?
c. What is the value of g (2, 10) ?
d. What are the range and domain ofg?
e. What is the value ofg(4, f (4))?
See all solutions

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