prove that the set of all languages that are not recursively enumerable is not countable.

Answers

Answer 1

To prove that the set of all languages that are not recursively enumerable is not countable, we need to use a proof by contradiction. Assume that the set of all languages that are not recursively enumerable is countable. This means that we can list all of these languages in a sequence, say L1, L2, L3, …, Ln.

Now, let's define a new language L that is not in this list. We construct L as follows: for every i-th language in the list, we choose a string that is not in Li and add it to L. Thus, L contains all strings that are not in any of the languages in the list.Next, we show that L is not recursively enumerable. Assume that L is recursively enumerable. Then there is a Turing machine that can enumerate all strings in L. Let's call this machine M. Since M is a Turing machine, we can construct a list of all strings that M generates, say s1, s2, s3, …. We note that every string in L is not in any of these strings, i.e., Li ∩ L = ∅ for all i.Now, let's construct a new string t as follows: for every i-th character in si, we choose a different character that is not equal to it. Thus, t is a string that is not generated by M, since it differs from every string generated by M in at least one position.However, t is in L, since it does not belong to any language in the list. This is a contradiction, since we assumed that M can enumerate all strings in L. Therefore, L is not recursively enumerable, and our assumption that the set of all languages that are not recursively enumerable is countable must be false.In conclusion, the set of all languages that are not recursively enumerable is uncountable.

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Answer 2

To prove that the set of all languages that are not recursively enumerable (also known as the set of co-recursively enumerable languages) is not countable, we can use a diagonalization argument.

Suppose, for the sake of contradiction, that the set of co-recursively enumerable languages is countable. Then we can list all of these languages as L1, L2, L3, ... in some fixed order.

Let M1 be a Turing machine that recognizes L1, and let M2 be a Turing machine that recognizes L2, and so on. We can assume that each Turing machine operates on a fixed alphabet, say Σ = {0, 1}, without loss of generality.

Now consider the language L = { w | w ∉ Lw }, where Lw is the language recognized by the Turing machine Mw. In other words, L is the language of all strings that are not in their corresponding language.

We claim that L is not in our list of co-recursively enumerable languages, which contradicts the assumption that our list is complete.

To see why L is not in the list, suppose for contradiction that there exists some Li such that L = Li. Then we know that there exists a Turing machine Mi that recognizes Li. But we can construct a string w such that w ∉ Li if and only if w ∈ L. Specifically, we can choose w to be the encoding of the Turing machine Mi itself, and then w ∈ L if and only if w ∉ Li. This contradicts the assumption that Li is co-recursively enumerable, since it implies that we can decide membership in Li by checking whether or not w is in L, which is impossible.

Therefore, we have shown that there exists a language L that is not in our list of co-recursively enumerable languages, which means that our list is incomplete. Since our list was assumed to be complete, we have reached a contradiction. Thus, the set of co-recursively enumerable languages is not countable.

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Related Questions

In this program you will draw some ASCII art using System.out.println(). Write a program that outputs exactly this drawing. ()_____ // O)___ // ) // _____/

Answers

This is known as ASCII Art. The program that give the above output is given below.

What is the program that give the above output?

The codes that resort to the above output are given below:


public class ASCIIArt{

public static void main(String[] args) {

System.out.println(" /)___");

System.out.println(" / o|___");

System.out.println(" / )");

System.out.println("/ )___/");

}

}

What is ASCII Art?

The 95 readable characters from the total of 128 established by the ASCII Standard from 1963 and ASCII compliant character sets with proprietary extra characters make up ASCII art, a visual design style that leverages computers for display (beyond the 128 characters of standard 7-bit ASCII).

The phrase is also figuratively used to describe all forms of text-based visual art. Any text editor may be used to make ASCII graphics, which is frequently used with free-form languages.

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Full Question:

See attached image.

For a direct mapped cache design with a 32-bit address, the following bits of the address are used to access the cache. assume a write through cache policy.
tag index offset
31-10 9-5 4-0
1. what is the cache block size (in words)?
2. 151 how many entries does the cache have?
3. 151 cod $5.3> what is the ratio between total bits required for such a cache implementation over the data storage bits?
address
starting from power on, the following byte-addressed cache references are recorded.
0 4 1 132 232 160 3024 30 140 3100 180 2180
4. [10 how many blocks are replaced?
5. what is the hit ratio?
6. list the final state of the cache, with each valid entry represented as a record of sindex, tag, data>

Answers

From the information given, the cache block size is 32. See explanation below.

How do we arrive at the cache block size?

Where the offset is 5 bits, the block size is given as:

2⁵ = 32.

Hence, the total number of blocks in the cache is 32.

How many entries does the cache have?

Note that

Total cache size = No. of entries (No. of tag bits + data bits + valid bit)

Hence,

= 32 x (22+ 256 +1)

= 8, 928 bits

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________ is installed in special, read-only memory in devices like printers or communication devices.

Answers

A firmware is installed in special, read-only memory in devices like printers or communication devices.

What is firmware and give example?

Firmware is known to be a kind of software that makes or produce the basic machine instructions that gives room for the hardware to act and be able to communicate with other kinds of software that is said to be running on a device.

Note that Firmware makes low-level control in for a lot of device's hardware and its example is microcontroller which is said to be an aspect of the microprocessor that informs the microprocessor the right actions to take.

Hence A firmware is installed in special, read-only memory in devices like printers or communication devices is the right answer.

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Every time you call a method, the address to which the program should return at the completion of the method is stored in a memory location called the ____.

Answers

Answer:

Stack

Explanation:

Every time you call a method, the address to which the program should return at the completion of the method is stored in a memory location called the stack.

Deploying an application across multiple __________________ can help build a robust architecture for an application that needs to have an up-time of at least 99.5

Answers

Deploying an application across multiple regions can help build a robust architecture for an application that needs to have an up-time of at least 99.5.

What is multi-region deployment?

The Multi-Region Infrastructure Deployment is known to be one that aids customers a lot to be able to easily control updates in regards to infrastructure needed for applications that are said to be deployed in all of primary and secondary Regions.

Hence, Deploying an application across multiple regions can help build a robust architecture for an application that needs to have an up-time of at least 99.5.

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What is the purpose of installing updates on your computer?

Answers

The purpose of adding updates to your computer are as follows:

Making sure your computer hardware can run the newest software programsEnsuring the built-in virus protection software is up to dateRemoving bugs and glitches from your operating systemAllowing your computer to run as fast as possible

Javascript and java are really just two slightly different names for the same language.
a) True
b) False

Answers

Answer:

False

Explanation:

JavaScript is language used along with html documents.

Java is a full-fledged programming language that handles applications.

find_cow(name, cows) Given a name and a Python list of Cow objects, return the Cow object with the specified name. If no such Cow object can be found, return None.

Answers

Using the computational knowledge in python it is possible to write a code that Given a name and a Python list of Cow objects

Writting the code in python:

def load_cows(filename):

     cow_dict = dict()

   f = open(filename, 'r')

   for line in f:

       line_data = line.split(',')

       cow_dict[line_data[0]] = int(line_data[1])

   return cow_dict

def greedy_cow_transport(cows, limit=10):

      trips = []

   cowsCopy = cows.copy()

   sortedCows = sorted(cowsCopy.items(), key=lambda x: x[1], reverse = True)

   while sum(cowsCopy.values()) > 0:

       ship = []

       total = 0

       for cow, value in sortedCows:

           if cowsCopy[cow] != 0 and value + total <= limit:

               ship.append(cow)

               total += value

               cowsCopy[cow] = 0

       trips.append(ship)

   return trips

def brute_force_cow_transport(cows,limit=10):

    trips = []

    possibilities = []

   for i in power_list:

       ship = []

       for j in i:

           ship_weights = []

           for k in j:

               ship_weights.append(cows[k])

               #print(ship_weights)

           ship.append(sum(ship_weights))

           #print(ship)

       if all(d <= limit for d in ship):

           possibilities.append(i)

     pruned_possibilities = []

   for k in possibilities:

       if k not in pruned_possibilities:

           pruned_possibilities.append(k)

   # now find the minimum list length:

   min_list_len = min(map(len, pruned_possibilities))

   for l in pruned_possibilities:

       if len(l) == min_list_len:

           return l

     

def compare_cow_transport_algorithms():

      greedy_start = time.time()

   greedy_results = greedy_cow_transport(cows, limit = 10)

   greedy_end = time.time()

   print('Greedy Algorithm time:', greedy_end -greedy_start)

   brute_force_start = time.time()

   brute_force_results = brute_force_cow_transport(cows, limit = 10)

   brute_force_end = time.time()

   print('Brute force time:', brute_force_end - brute_force_start)

   print('Greedy Algorithm results:', greedy_results)

   print('Number of trips returned by Greedy Algorithm:', len(greedy_results))

   print('Brute Force Algorithm results:', brute_force_results)

   print('Number of trips returned by Brute Force Algorithm:', len(brute_force_results))

   

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What name is given to people who break into computer systems with the sole purpose to steal or destroy data?

Answers

Answer:

those people are called hackers

Explanation:

Answer:

They are called hackers.

Explanation:

hope it helps

At startup, Windows Server 2012 Core presents _______.

Answers

At startup, Windows Server 2012 Core presents a command window.

What is a Windows Server used for?

Windows Server is known to be a kind of a group of operating systems that is made by Microsoft that aids enterprise-level management, data storage, applications, and others.

The Command window is known to be that which gives a person  the information, it also gives instructions, and gives room for one  to be able to enter commands usually at the Command prompt.

Therefore, note that at startup, Windows Server 2012 Core presents a command window as it is the first thing that will come up.

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You need a(n) _____ to play an mp3 audio file on a desktop or laptop computer

Answers

You need a stand alone player to play an mp3 audio file on a desktop or laptop computer.

What is a stand alone player?

The standalone player is known to be a device that gives room for a person or is one that allows a Blender game to be able to run without one trying to load the Blender system.

Note that in the above, You need a stand alone player to play an mp3 audio file on a desktop or laptop computer.

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Joe is examining the logs for his web server and discovers that a user sent input to a web application that contained the string WAITFOR. What type of attack was the user likely attempting

Answers

SQL Injection Attack

Choose the term that matches each definition.

: increasing capacity by adding nodes or units


: increasing capacity by upgrading individual resources to be more powerful

Horizontal Scaling
Vertical Scaling

Answers

Horizontal Scaling is the act of increasing capacity by adding nodes or units while increasing capacity by upgrading individual resources to be more powerful is called Vertical Scaling.

What is vertical and horizontal scalability?

Horizontal scaling is known to be the act of  adding more nodes while vertical scaling is a term that connote the act of adding a lot of power to one's current machines.

Therefore, Horizontal Scaling is the act of increasing capacity by adding nodes or units while increasing capacity by upgrading individual resources to be more powerful is called Vertical Scaling.

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An operator can be overloaded to define functionality that is drastically different from the original operator.
a. True
b. False

Answers

A. True

Explanations:

You can redefine or overload the function of most built-in operators in C++. These operators can be overloaded globally or on a class-by-class basis. Overloaded operators are implemented as functions and can be member functions or global functions. An overloaded operator is called an operator function.

Mary and Billy are moving away to college and both know in their hearts that they need to end their relationship. They move away with a mutual agreement to breakup and leave peacefully. What is this called

Answers

In the case of Mary and Billy, the above is called Negotiated farewell.

What is Negotiated farewell?

In the case of  Negotiated Farewell, one can say that couples are knonw to jointly create the story of the end to what we say their marriage.

Note that Negotiated Farewell is one that is made for emotionally immature or any kind of combative people. hence, In the case of Mary and Billy, the above is called Negotiated farewell.

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Which epolicy orchestrator component gathers the events from the managed systems and uploads them to the epo server?

Answers

The policy orchestrator component that gathers the events from the managed systems and uploads them to the epo server is mcAfee agent.

Which ePolicy orchestrator component  is enforced?

The McAfee Agent can be described as the one that is distributed component of McAfee ePolicy Orchestrator (McAfee ePO).

It helps in downloads and enforcement of policies and it executes client-side tasks.

To find  find  McAfee agent, one can follow these steps:

View McAfee Agent Status MonitorOn the managed system,  select McAfee Agent Status Monitor. Save Contents to Desktop to save the contents of the McAfee Agent log to a file.

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If a DBMS enforces a DELETE CASCADE option on the referential integrity constraint between SELLER and REALTOR in the HOMETOWN REALESTATE database, what will be the outcome after a user tries to delete the third record (R3, Cliff) from REALTOR

Answers

The outcome will be option b: ) CLIENT will have 6 records, AGENT will have 3 records.

What are client records?

Client record is known to be a form of a written composition of information that tells and documents the examination , diagnosis or others of a client.

Based on the above scenario, The outcome will be option b: ) CLIENT will have 6 records, AGENT will have 3 records.

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When you group together related variables, the group is referred to as a(n) ____ of variables

Answers

Answer:

Array of variables

Explanation:

This occurs when you group together related variables

Given five positive integers, find the minimum and maximum values that can be calculated by summing exactly four of the five integers. Then print the respective minimum and maximum values as a single line of two space-separated long integers.

Answers

Using the knowledge in computational language in JAVA it is possible to write a code that  print the respective minimum and maximum values as a single line

Writting the code in JAVA:

import java.io.*;

import java.util.*;

import java.text.*;

import java.math.*;

import java.util.regex.*;

public class Solution {

   public static void main(String[] args) {

       Scanner in = new Scanner(System.in);

       long sum = 0;

       long max = Long.MIN_VALUE;

       long min = Long.MAX_VALUE;

       for (int i = 0; i < 5; i++){

           long n = in.nextLong();

           sum += n;

           max = Math.max(max, n);

           min = Math.min(min, n);

       }

       System.out.println((sum - max) + " " + (sum - min));

   }

}

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What are the three different types of hard drives? Briefly describe each, and list the possible configurations and device files for each type. Describe RAID level 5. What are the advantages of using RAID level 5 over other types of RAID?

Answers

The three different types of hard drives are:

SATASSD NVMe

What is the SATA drive?

This was said to be Introduced in the year 2003, SATA (or Serial Advanced Technology Attachment) is known to be a kind of a default interface that was made for a lot of desktop and laptop hard drives.

Hence, The three different types of hard drives are:

SATASSD NVMe

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Any two differences between second generation and third generation Write any generation computer.​

Answers

Answer:


The second generation computer is based on the transistor whereas the third generation computer is based on the integrated chip.

Explanation:

Anonymity on the internet has lowered the cost of rudely confronting people. what has happened to the supply of rude confrontations?

Answers

The thing that has happened to the supply of rude confrontations is that  The supply has increased, shifting down and to the right.

What supply means in economics?

Supply in economics is known to be seen as the full amount of a given product or service that a given  supplier is known to offers to consumers at a specific point in time and a given price level.

Hence, The thing that has happened to the supply of rude confrontations is that  The supply has increased, shifting down and to the right because Anonymity on the internet has lowered the cost of rudely confronting people.

See full question below

Anonymity on the Internet has lowered the cost of rudely confronting people. What has happened to the supply of rude confrontations? a) The supply has decreased, shifting up and to the left. b) The supply has increased, shifting up and to the left. C) The supply has decreased, shifting down and to the right. d) The supply has increased, shifting down and to the right. Question 46 (1 point) Figure: Supply Shift Saved Price $100-

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In which directory would a system administrator store scripts that should be run monthly by the cron daemon?

Answers

Answer: /etc/cron.monthly

Explanation:

Every time a user enters a different web page url into a browser, the same web page appears. what will help in this situation?atask managerbfirewallcanti-malwareddevice manager

Answers

Every time a user enters a different web page url into a browser, the same web page appears. the issue that can help is the use of firewall.

What is firewall and how it works?

The term firewall is known to be that which helps protect a person's network from attackers.

Note that A firewall shields one's network because it work in a 24/7 filter window, and this is one that often works by scanning the data that tries to enter your network and hinder anything that may act or looks suspicious from getting through.

Hence, in the case above, Every time a user enters a different web page url into a browser, the same web page appears. the issue that can help is the use of firewall.

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"necessarily is the mother of computer " justify this statement with respect to the evolution of computer.​

Answers

Explanation:

Computers in the form of personal desktop computers, laptops and tablets have become such an important part of everyday living that it can be difficult to remember a time when they did not exist. In reality, computers as they are known and used today are still relatively new. Although computers have technically been in use since the abacus approximately 5000 years ago,

What are the nuclear codes?

Answers

Answer:

Nuclear codes allow the president to prove the individual is authorized to order the launch.

Explanation:

Hope this helps!

If not, I am sorry.

The security team at an organization looks to protect highly confidential servers. Which method does the team propose when protecting the servers against explosives?

Answers

The method that the team propose in protecting the servers against explosives is Physical Security.

What are the Physical Security Considerations?

Physical security is known to be a key part as it acts in network security and tends to protect an organization and employees.

Note that   An organization's physical components is one that can have vulnerabilities that need to be mitigated by using appropriate physical security measures by:

The use of Building and grounds control Fire risks control, etc.

Hence, The method that the team propose in protecting the servers against explosives is Physical Security.

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Simple statistics looks catalog 3.15 LAB: Simple statistics Given 4 floating-point numbers. Use a string formatting expression with conversion specifiers to output their product and the average as integers (rounded, then as floating point numbers Output each rounded integer using the following print (1.0).tornat (your_value)) Output each floating point value with three digits after the decimal point, which can be achieved as follows print(1.3).format(your value)) Ex If the input is 3.3 10. 5.0 the output is 20722 2071. 7. 125 ACTIVITY 3151 LAB Simple statistics 0/10 main.py 1 2 - Floatinet()) - Floatinus Fotout Flatirot)

Answers

Based on the given instructions:

1. you must set the display format for all values: %0.2f

2. use f-string in this format: f"{}" and add your value in {}

3. int: convert float to integer and round: round up the number

The given code

num1 = float(input())

num2 = float(input())

num3 = float(input())

average = (num1+num2+num3) / 3

your_value = num1*num2*num3

print('%0.0f %0.0f %0.2f %0.2f' % (int(average), int(your_value), average, your_value))

# or:

print(f"{int(average)} {int(your_value)} {round(average,2)} {round(your_value,2)}")

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connection failed while opening file within cryptographic module - mbedtls_ssl_handshake returned -9984 ( X509 - Certificate verification failed, e.g. CRL, CA or signature check failed )

Answers

The connection that failed while opening file within cryptographic module - mbedtls_ssl_handshake returned -9984 ( X509 - Certificate verification failed, e.g. CRL, CA or signature check failed is known to be a kind of a TLS hanshake failing with EC.

What are cryptographic products?

These are known to be Software, hardware or what we call the firmware that is made up of one or a lot of cryptographic functions.

Note that a cryptographic product is one that is made up of a kind of cryptographic module and they also help to implements security functions (such as cryptographic algorithms),

Therefore, based on the above, The connection that failed while opening file within cryptographic module - mbedtls_ssl_handshake returned -9984 ( X509 - Certificate verification failed, e.g. CRL, CA or signature check failed is known to be a kind of a TLS hanshake failing with EC.

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Write a short program that asks the user to enter a month & prints a message based on the month

Answers

Answer:

The Code:

def month(x):

    if (x==1):

        print ("January")

    if (x==2):

        print("February")

    if (x==3):

        print("March")

    if (x==4):

        print("April")

    if (x==5):

        print("May")

    if (x==6):

        print("June")

    if (x==7):

        print("July")

    if (x==8):

        print("August")

    if(x==9):

        print("September")

    if(x==10):

        print("October")

    if(x==11):

        print("November")

    if(x==12):

        print("December")

    if(x<1 or x>12):

        print("Wrong Input! Try again")

month = int(input("Enter the month number: "))

month(month)

Explanation:

The above program is written in the PYTHON programming language.

In this program, the user inputs the number of any month, showing the month name in the output. For example,

if user enters 1, month name = JANUARY

if user enters 5, month name = MAY

and if the user enters 13, then output = Wrong Input! try again

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Why do you choose study in ALS find y'. y = log6(x4 5x3 2) a does anyone know it Find a formula for the exponential function passing through the points ( -3, 3 /25) and (1,15) f(X) = _______- Domino's growth plan has focused on Mexico partially because ___ True/False: for selling in relationship-oriented countries, a sales force consisting of american expatriates proves to be most efficient. The Biology Department offers an upper-level elective called "Biogeography Which of the following topics would you least expect to be taught in this course? a.How plate tectonics and the separation of continents influenced movements of organisers over geological time b.The current distribution patterns of certain species across the globe. c.The influence of abiotic factors on community structure d.The specific composition of rocks in a particular ecosystem Based on your data, about the simple cell, what can you conclude about a horizontal stimulus?a. it is inhibitory b. it is somewhat excitatory c. it is very excitatory Compare the diffusion coefficients of carbon in BCC and FCC iron at the allotropic transformation temperature of 912C and explain the reason for the difference in their values. upiter Explorers has $5,600 in sales. The profit margin is 3 percent. There are 4,000 shares of stock outstanding, with a price of $1.20 per share. What is the company's price-earnings ratio? Multiple Choice 6.12 times 5.04 times 14.29 times 28.57 times 28.00 times Analyze Felipe's work. Is he correct? No, he did not substitute into the formula correctly in step 1 No, he incorrectly evaluated the powers in step 2.No, he did not add correctly in step 3. Yes, he calculated the distance correctly. Lead-210 is a radioactive isotope that sometimes decays through this pathway: B, a, b, b, b, a. What nuclides form in this decay series? nuclide after the first decay: ..............nuclide after the second decay: ...............nuclide after the third decay: ..................nuclide after the fourth decay:...............nuclide after the fifth decay: ..............nuclide after the sixth decay:................ a treatment for phobias and other anxiety disorders, involves re-pairing a feared object with a new response that is incompatible with anxiety. group of answer choices Eli Whitney saw replicas of his cotton gin at work on countless southern plantations. However, they were built without honoring the patent he held on the device that is, without his making any profit on this profitable idea. This prompted Whitney to remark that, "an invention can be so valuable as to be worthless to the inventor."Learning from this experience, Whitney turned his attention away from inventions and to his concept of mass production. In 1797, the government was worried about possible war with France. Forty thousand more muskets were needed than the1,000-per-three-years produced by the national armory. Whitney reasoned that musket production was so slow because each musket was handmade, unique, and therefore impossible to construct or repair, except with parts specially made for each unit.The problem could be solved, he believed, if all the musket parts could be made uniform and manufactured by machine tools which could be operated by unskilled laborers. Then all the parts would be interchangeable, and anyone could assemble (as well as repair) a musket from a large supply of parts.With this idea in mind, he proposed delivering 10,000 muskets in two years. However, the time it would take to deliver the 10,000 muskets would stretch from his estimated two years to more than ten this due to circumstances not connected to his proposed method. But in 1801, Whitney was able to prove his point in front of President-elect Thomas Jefferson and other officials, when those present randomly selected parts from dis-assembled muskets and, from them, assembled complete muskets. It was the beginning of the American system of mass production.What is the author's tone in this piece? Find the length of the curve.r(t) =leftangle2.gif6t, t2,19t3rightangle2.gif, 1. In Mathevon et al. (2010) study of hyena laughter, or "giggling", they asked whether sound spectral properties of hyena's giggles are associated with age. The data show the giggle frequency (in hertz) and the age (in years) of 16 hyena. Age (years) 2 2 2 6 9 10 13 10 14 14 12 7 11 11 14 20 Fundamental frequency (Hz) 840 670 580 470 540 660 510 520 500 480 400 650 460 500 580 500 (a) What is the correlation coefficient r in the data? (Follow the following steps for your calculations) (i) Calculate the sum of squares of age. (i) Calculate the sum of squares for fundamental frequency. (iii) Calculate the sum of products between age and frequency. (iv) Compute the correlation coefficient, r. you are the manager of a monopoly that faces an inverse demand curve described by p = 200 15q. your costs are c = 15 20q. the profit-maximizing price is: TRUE OR FALSE the strategies used for supported employment are not also used when providing educational services Statement 1: There are three Stooges: Moe, the leader, Larry, the follower, and Curly, the fool. Statement 2: The best choices for dessert are cupcakes, pudding, and truffles. Statement 3: There are four Marx Brothers: Harpo, Chico, Zeppo, and Groucho. Statement 4: Upon reflection I have discovered these facts: Tom is cruel, Jerry is silly, and Mom is mad. 1) Which of these statements requires semicolons to separate the items in the series?A) Statement 1B) Statement 2C) Statement 3D) Statement 4 Score for Weston 7. Of 20 points)1. Johnny printed two maps of a walking trail near his home. The length of the walking trail onthe first map is 8 cm,A. Choose a length between 5cm and 15cm for the walking trail on the second map. cm. (1 point)B. Determine the scale factor from the first map to the second map. Show your work. Itmay help you to "draw" out the two maps and label the trails and their correspondinglengths. (2 points)newScale factor =oldC. A landmark on the first map is a triangle with side lengths of 3 mm, 4 mm, and 5 mm. What are the side lengths of the triangle landmark on the second map? Show yourwork. It may help you to draw out the two maps and label the trails. (3 points)