![Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991](https://www.bartleby.com/isbn_cover_images/9781259731709/9781259731709_smallCoverImage.gif)
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
8th Edition
ISBN: 9781259731709
Author: ROSEN
Publisher: MCG
expand_more
expand_more
format_list_bulleted
Textbook Question
Chapter 7.3, Problem 18E
18.Suppose that a Bayesian spam filter is trained on a set of 500 spam messages and 200 messages that are not spam, The word “exciting'' appears in 40 spam messages and in 25 messages that are not spam. Would an incoming message be rejected as spam if it contains the word “exciting1and the threshold for rejecting spam is 0.9?
Expert Solution & Answer
![Check Mark](/static/check-mark.png)
Want to see the full answer?
Check out a sample textbook solution![Blurred answer](/static/blurred-answer.jpg)
Students have asked these similar questions
The five most common words appearing in spam emails are shipping!, today!, here!,available, and fingertips! (Andy Greenberg, “The Most Common Words In Spam Email,”Forbes website, March 17, 2010). Many spam filters separate spam from ham (email notconsidered to be spam) through application of Bayes’ theorem. Suppose that for one emailaccount, 1 in every 10 messages is spam and the proportions of spam messages that havethe five most common words in spam email are given below.shipping! .051today! .045here! .034available .014fingertips! .014Also suppose that the proportions of ham messages that have these words areshipping! .0015today! .0022here! .0022available .0041fingertips! .0011a. If a message includes the word shipping!, what is the probability the message is spam?If a message includes the word shipping!, what is the probability the message is ham?Should messages that include the word shipping! be flagged as spam?b. If a message includes the word today!, what is the probability the…
The five most common words appearing in spam emails are shipping!, today!, here!,available, and fingertips! (Andy Greenberg, “The Most Common Words In Spam Email,”Forbes website, March 17, 2010). Many spam filters separate spam from ham (email notconsidered to be spam) through application of bayes’ theorem. Suppose that for one emailaccount, 1 in every 10 messages is spam and the proportions of spam messages that havethe five most common words in spam email are given below.shipping! .051today! .045here! .034available .014fingertips! .014Also suppose that the proportions of ham messages that have these words areshipping! .0015today! .0022here! .0022available .0041fingertips! .0011a. If a message includes the word shipping!, what is the probability the message is spam?If a message includes the word shipping!, what is the probability the message is ham?Should messages that include the word shipping! be flagged as spam?b. If a message includes the word today!, what is the probability the…
The five most common words appearing in spam emails are shipping!, today!, here!,available, and fingertips! (Andy Greenberg, “The Most Common Words In Spam Email,”Forbes website, March 17, 2010). Many spam filters separate spam from ham (email notconsidered to be spam) through application of bayes’ theorem. Suppose that for one emailaccount, 1 in every 10 messages is spam and the proportions of spam messages that havethe five most common words in spam email are given below.shipping! .051today! .045here! .034available .014fingertips! .014Also suppose that the proportions of ham messages that have these words areshipping! .0015today! .0022here! .0022available .0041fingertips! .0011a. If a message includes the word shipping!, what is the probability the message is spam?If a message includes the word shipping!, what is the probability the message is ham?Should messages that include the word shipping! be flagged as spam?
Chapter 7 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
Ch. 7.1 - i. What is the probability that a card selected at...Ch. 7.1 - t istheprobability that a fair die comes up six...Ch. 7.1 - t is the probability that a randomly selected...Ch. 7.1 - What is the probability7that a randomly selected...Ch. 7.1 - t is the probability that the sum of the numbers...Ch. 7.1 - t is the probability that a card selected at...Ch. 7.1 - t is the probability that when a coin is flipped...Ch. 7.1 - t is the probability that a five-card poker hand...Ch. 7.1 - t is the probability that a five-card poker hand...Ch. 7.1 - t is the probability that a five-card poker hand...
Ch. 7.1 - Prob. 11ECh. 7.1 - t is the probability that afive-card poker hand...Ch. 7.1 - t is the probability tliat afive-card poker hand...Ch. 7.1 - t istheprobability that a five-card poker hand...Ch. 7.1 - t is theprobabilifrthatafive-cardpoker hand...Ch. 7.1 - t is the probability7that a five-card poker hand...Ch. 7.1 - Prob. 17ECh. 7.1 - Mat is the probability' that a five-card poker...Ch. 7.1 - Prob. 19ECh. 7.1 - probabihh’thatafiM^Ch. 7.1 - Prob. 21ECh. 7.1 - t is the probability that a positive integer not...Ch. 7.1 - t is the probability that a positive integer not...Ch. 7.1 - Prob. 24ECh. 7.1 - - Find the probability of winning a lottery by...Ch. 7.1 - 26.Find the pr obabilitj- of selecting none of the...Ch. 7.1 - Prob. 27ECh. 7.1 - Prob. 28ECh. 7.1 - Prob. 29ECh. 7.1 - Prob. 30ECh. 7.1 - Prob. 31ECh. 7.1 - Prob. 32ECh. 7.1 - i$theprobabilitytiiatAbby,Barry,andSy^...Ch. 7.1 - 34.Mat is the probability' that Bo, Colleen, Jeff,...Ch. 7.1 - roulette, a wheel with 38 numbers is spun. Of...Ch. 7.1 - ch is more likely: rolling a total of 8 when two...Ch. 7.1 - ch is more likely: rolling a total of 9 when hvo...Ch. 7.1 - A player in the Mega Millions lottery picks five...Ch. 7.1 - a player buys a Mega Millions ticket in many...Ch. 7.1 - A player in the Powerball lottery picks five...Ch. 7.1 - Aplayer in the Powerball lottery (see Exercise 40)...Ch. 7.1 - Two events E i and E2are calledindependentifp(Etfl...Ch. 7.1 - Prob. 43ECh. 7.1 - Suppose that instead of three doors, there are...Ch. 7.1 - s problem was posed by the Chevalier de Mere and...Ch. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - w that conditions (2) and (22) are met under...Ch. 7.2 - A pair of dice is loaded. The probability that a 4...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - 8.What is the probability of these events when we...Ch. 7.2 - t is the probability of these events when we...Ch. 7.2 - What is the probability of these events when we...Ch. 7.2 - pose, that £ and F are. events such that d(£)=0.7...Ch. 7.2 - pose that £ and Fare events such thatp(£) = 0.8...Ch. 7.2 - w that if £ and F are events, thenpfEn F) >p(E) +...Ch. 7.2 - Use mathematical induction to prove the following...Ch. 7.2 - w that if £x, £2,Enare events from afinite sample...Ch. 7.2 - Show that iff and f are independent events,...Ch. 7.2 - 17,It £ and F are independent events, prove or...Ch. 7.2 - What is the probability that hvo people chosen at...Ch. 7.2 - Mat is the probability that two people chosen at...Ch. 7.2 - Prob. 20ECh. 7.2 - Prob. 21ECh. 7.2 - February 29 occurs only inleap years, Years...Ch. 7.2 - ^Tiat is the conditional probabilitv that exactly...Ch. 7.2 - What is the. conditional probabilih' that exactly...Ch. 7.2 - Prob. 25ECh. 7.2 - Let Ebe the event that aranmly generated bit...Ch. 7.2 - Prob. 27ECh. 7.2 - a8. Assume that the probability a child is a boy...Ch. 7.2 - A group of six people play the game of “ odd...Ch. 7.2 - Find the probability that a randomly generated bit...Ch. 7.2 - Find the probability that a family with five...Ch. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Find each of the following probabilities...Ch. 7.2 - d each of the following probabilities...Ch. 7.2 - Prob. 36ECh. 7.2 - Prob. 37ECh. 7.2 - 38.A pair of dice is rolled in a remote location...Ch. 7.2 - This exercise employs the probabilistic method to...Ch. 7.2 - Dense a Monte Carlo algorithm that determines...Ch. 7.2 - pseudocode to write out the probabilistic...Ch. 7.3 - i.Suppose that £ andFare events in a sample space...Ch. 7.3 - Suppose that Land Fare events in a sample space...Ch. 7.3 - 3.Suppose that Frida selects a ball by first...Ch. 7.3 - 4.Suppo s e that Ann selects a ball by first...Ch. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - 8,Suppose that one person in 10,000 people has a...Ch. 7.3 - Suppose that 8% of the patients tested in a clinic...Ch. 7.3 - io,Suppose that 4% of the patients tested in a...Ch. 7.3 - ...Ch. 7.3 - ...Ch. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - In this exercise we will use Bayes' theorem to...Ch. 7.3 - Prob. 16ECh. 7.3 - Prob. 17ECh. 7.3 - 18.Suppose that a Bayesian spam filter is trained...Ch. 7.3 - 19,Suppose that a Bayesian spam filter is trained...Ch. 7.3 - Prob. 20ECh. 7.3 - ,Suppose that a Bayesian spam filter is trained on...Ch. 7.3 - Suppose that we have prior information concerning...Ch. 7.3 - Prob. 23ECh. 7.4 - t is the expected number of heads that come up...Ch. 7.4 - t is the expected number of heads that come up...Ch. 7.4 - t is the expected number of times a 6 appears when...Ch. 7.4 - A coin is biased so that the probability a head...Ch. 7.4 - ^Tiat is the expected sum of the numbers that...Ch. 7.4 - Prob. 6ECh. 7.4 - final exam of a discrete mathematics course...Ch. 7.4 - t is the expected sum of the numbers that appear...Ch. 7.4 - Prob. 9ECh. 7.4 - Suppose that we flip a fair coin until either it...Ch. 7.4 - Suppose that we roll a fair die until a 6 conies...Ch. 7.4 - pose that we roll a fair die until a 6 comes up....Ch. 7.4 - pose thatwerollapairoffair dice...Ch. 7.4 - Show that the sum of the probabilities of a random...Ch. 7.4 - Show that if the random variable A'has the...Ch. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Show that if J2,...,Xnare mutually independent...Ch. 7.4 - What is the expected value of the sum of the...Ch. 7.4 - as.Provethelaw of total expectations.Ch. 7.4 - Prob. 23ECh. 7.4 - Prob. 24ECh. 7.4 - A run is a maximal sequence of successes in a...Ch. 7.4 - a6.Let J(s) be a random variable, where I(s) is a...Ch. 7.4 - What is the variance of the number of heads that...Ch. 7.4 - t is the variance ot the number of times a 6...Ch. 7.4 - LetXnbe the random variable that equals the number...Ch. 7.4 - w that ifXand Fare independent random variables,...Ch. 7.4 - Prob. 31ECh. 7.4 - Pronde an example that shows that the variance of...Ch. 7.4 - pose that A\ andX2are independent Bernoulli trials...Ch. 7.4 - Prove the general caseofTheoremy. That is, show...Ch. 7.4 - Prob. 35ECh. 7.4 - Prob. 36ECh. 7.4 - Prob. 37ECh. 7.4 - pose that the number of cans of soda pop filled in...Ch. 7.4 - 39.Suppose that the number of aluminum cans...Ch. 7.4 - pose the probabilitvthatxis the...Ch. 7.4 - In this exercise we derive an estimate of the...Ch. 7.4 - Prob. 42ECh. 7.4 - to is the variance of the number of fixed...Ch. 7.4 - Prob. 44ECh. 7.4 - Prob. 45ECh. 7.4 - Prob. 46ECh. 7.4 - Prob. 47ECh. 7.4 - Prob. 48ECh. 7.4 - Prob. 49ECh. 7 - Define the probability of an event when all...Ch. 7 - WTiat conditions should be met by the...Ch. 7 - Define, the conditional probability’ of an event £...Ch. 7 - Prob. 4RQCh. 7 - tois a random variable? toare the possible values...Ch. 7 - Prob. 6RQCh. 7 - Explain how the average-case computational...Ch. 7 - Prob. 8RQCh. 7 - What does the linearity of expectations of random...Ch. 7 - Prob. 10RQCh. 7 - Prob. 11RQCh. 7 - Prob. 12RQCh. 7 - Prob. 13RQCh. 7 - What is the variance of the sum of n independent...Ch. 7 - Prob. 15RQCh. 7 - Prob. 1SECh. 7 - 2012, a player in the Mega Millions lottery picks...Ch. 7 - 2012, a player in the Powerball lottery picks five...Ch. 7 - t is the probability that a hand of 13 cards...Ch. 7 - t is the probability that a 13-card bridge hand...Ch. 7 - t is the probability that a seven-card poker hand...Ch. 7 - What is the expected value of the number that...Ch. 7 - What is the expected value of the number that...Ch. 7 - Suppose that a pair of fair octahedral dice is...Ch. 7 - io.Suppose that a pair offaiir dodecahedral diceis...Ch. 7 - ii.Supp o s e that a fair standard (cubic) die and...Ch. 7 - Prob. 12SECh. 7 - (mpeople!n>3!play“oddp™ut’todeadeMo^...Ch. 7 - Prob. 14SECh. 7 - posethatmandnarepositiYeintegers.Bat is...Ch. 7 - pose thatEt, E2,Enarenevents with p(£j) >o fori...Ch. 7 - Prob. 17SECh. 7 - t is the probability that when a fair coin is...Ch. 7 - t is the probability that a randomly selected bit...Ch. 7 - t is the probability that a randomly selected bit...Ch. 7 - sider the following game. A per son flips a coin...Ch. 7 - pose that n halls are tossed intobbins so that...Ch. 7 - posethatAandBareeventswthprobabilitiesp(A) =...Ch. 7 - posethat/l andB are events...Ch. 7 - all fromDefinition jinSection 7,2that the events...Ch. 7 - ...Ch. 7 - Prob. 27SECh. 7 - Prob. 28SECh. 7 - Prob. 29SECh. 7 - Prob. 30SECh. 7 - Prob. 31SECh. 7 - Prob. 32SECh. 7 - Prob. 33SECh. 7 - maximum satisfiability problemasks for an...Ch. 7 - Prob. 35SECh. 7 - The following method can be used to generate a...Ch. 7 - Prob. 1CPCh. 7 - Prob. 2CPCh. 7 - Prob. 3CPCh. 7 - Prob. 4CPCh. 7 - Prob. 5CPCh. 7 - ...Ch. 7 - Prob. 7CPCh. 7 - Prob. 8CPCh. 7 - Prob. 9CPCh. 7 - ulaterepeated trials oftheMoufr Hall Three-Door...Ch. 7 - Prob. 11CPCh. 7 - Prob. 1CAECh. 7 - Prob. 2CAECh. 7 - Prob. 3CAECh. 7 - Prob. 4CAECh. 7 - Prob. 5CAECh. 7 - Prob. 6CAECh. 7 - Prob. 7CAECh. 7 - Prob. 8CAECh. 7 - cribe the origins of probability theory and the...Ch. 7 - Prob. 2WPCh. 7 - 3.Discuss the probability' of winning when you...Ch. 7 - estigate the game of craps and discuss the...Ch. 7 - Prob. 5WPCh. 7 - Prob. 6WPCh. 7 - lain how Erdos and Renvi first used the...Ch. 7 - cuss the different types of probabilistic...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Suppose the probability of incorrectly transmitting a single bit is . Compute the probability of correctly receiving a -word coded message made up of -bit words.arrow_forwardThe five most common words appearing in spam emails are shipping!, today!, here!,available, and fingertips! (Andy Greenberg, “The Most Common Words In Spam Email,”Forbes website, March 17, 2010). Many spam filters separate spam from ham (email notconsidered to be spam) through application of bayes’ theorem. Suppose that for one emailaccount, 1 in every 10 messages is spam and the proportions of spam messages that havethe five most common words in spam email are given below.shipping! .051today! .045here! .034available .014fingertips! .014Also suppose that the proportions of ham messages that have these words areshipping! .0015today! .0022here! .0022available .0041fingertips! .0011a. If a message includes the word shipping!, what is the probability the message is spam?If a message includes the word shipping!, what is the probability the message is ham?Should messages that include the word shipping! be flagged as spam?b. If a message includes the word today!, what is the probability the…arrow_forward15. A marketing firm is hired to conduct research into the listening habits of drivers in u large urban arca. On the first day, 121 drivers were surveyed: 26 listen to FM radio while driving, 4 of whom listen to only FM. Eight more listen to FM and AM only, while 4 listen to FM and satellite only. There were 6 that listen to only AM, 22 that listen to only satellite, and 69 that listen to at least one of the three. (a) Are there more people who listen to satellite radio, or more who listen to none of the three types in the survey? (b) How many more listen to AM radio than FM? (c) How many listen to some form of radio, but not AM?arrow_forward
- 8. Cialdini (2003) investigated how both social norms impacted people's behaviors. Using his work as a means to decrease screen time (injunctive norm), student researchers manipulated a comparison group (U.S. and Global reference groups) and an injunctive norm (descriptive norm inconsistent with injunctive people continuing high use of screen time versus descriptive norm consistent with injunctive = people trying to decrease screen time). After reading about screen time and what various people's screentime behaviors were, participants indicated their intention for future screen time. Did the social norms or comparison groups impact future intended screen time? Fully test the research question (p<.01) and indicate effect sizes. U.S. Reference Group = Global Reference Group G=227.31 N=84 Descriptive Norm Inconsistent n = 22 M= 2.63 T = 57.86 SS = 35.49 n=21 M = 2.38 T = 49.98 SS = 30.41 Descriptive Norm Consistent n = 20 M=2.75 T = 55.00 SS = 94.06 n=21 M=3.07 T=64.47 SS = 26.08arrow_forwardSuppose that a Bayesian spam filter is trained on a set of 5000 spam messages and 7500 messages that are not spam. The word "achievement" appears in 1000 spam messages and 150 messages that are not spam, while the word “research" appears in 800 spam messages and 300 messages that are not spam. Estimate the probability that a received message containing both the words "achievement" and "research" is spam. Will the message be rejected as spam if the threshold for rejecting spam is 0.9?arrow_forwardA company wants to learn the customers’ purchase habits of buying certain products. This company did a survey about a random 1000 customers, who were asked about the times they purchased the certain products within one month. Then what is the data in this example (choose the answer that contains all the information received): -All the records times of purchasing the certain products for all of the 1000 customers. -Customers’ purchase habits. -Times the customers purchased the certain products. -1000 random customers.arrow_forward
- An experiment was conducted to determine whether a test designed to identify a certain formof mental illness could be easily interpreted with little psychological training. The test was givento 100 people (half of which had the illness, and half did not) and fifteen people were asked toevaluate them. The fifteen judges were five staff members of a mental hospital, five trainees atthe hospital, and five undergraduate psychology majors. The results in the table give the numberof the 100 tests correctly classified by each judge. Analyze the data with the Kruskal-Wallis orFriedman test statistics.Staff Trainees Students78 80 6576 69 7480 75 7879 81 8086 72 75a) Is there evidence to suggest that there are differences between staff, trainees, and…arrow_forwardA pet store conducted a survey to gather information about the types of pets people own. Here are the results. • 603 people were surveyed in all • 297 of the people own a bird 62 of the people own both a bird and a fish, but don't own a cat • 29 of the people own both a bird and a cat, but don't own a fish • 47 of the people own a bird, but own neither a fish nor a cat • 117 of the people own a cat, but own neither a bird nor a fish • 56 of the people own a fish, but own neither a bird nor a cat • 572 of the people own at least one of these three types of pets (bird, fish, cat) Construct a Venn diagram illustrating these results. Then answer the question. All people in the survey How many of the people don't own a cat? | people Own a bird Own a fish Own a cat Continue O 2021 McGraw Hill LLC. Al Rights Reserved. Terms of Use I F 3°C Nub para buscar 口 口 口arrow_forward1) A doctor wishes to know if coffee consumption by coffee drinkers has changed in the last year. She surveys 400 patients from her practice who are known coffee drinkers. She found that 312 patients drink more than 2 cups of coffee per day. The rest of the respondents drink only 1 – 2 cups of coffee per day. The doctor’s previous research found that 75% of coffee drinkers drink more than 2 cups of coffee per day. a) What are the null and alternative hypotheses for this study? Is this a one-tailed or a two-tailed test? b) What test should the doctor use to test her hypothesis? Justify your answer. c) Based in your answers to Parts A and B above, calculate your appropriate test statistic. d) Assume that the doctor set α to 0.05 during the design phase of her study. -Calculate the degrees of freedom. - State final conclusion for this study based on α and the degrees of freedom.arrow_forward
- A) one sample of n=10 scores has a mean of m=8, and a second sample of n=5 scores has a mean of m=12. If the two samples are combined, what is the mean for the combined sample? One sample has a mean m=6, and a second sample has a mean of m=12, the two samples are combined into a single set of scores. B) what is the mean for the combined set if both of the orginal samples has n=4? C) what is the mean for the combined set it the first sample has n=3 and second sample has n=6?arrow_forwardAccording to the Max Markwell classifica- tion, there are three types of drivers on High- way 71 between Bastrop, Texas and Austin, Texas: Good, Bad, and Egregious. 12% of all drivers are Good ●68% of all drivers are Bad • All remaining drivers are Egregious Good drivers drive in the passing lane 28% of the time (e.g., only while actively passing or making a left turn within the next 1/2 mile) • Bad drivers drive in the passing lane 74% of the time Egregious drivers camp in the passing lane 100% of the time. You observe a driver in the passing lane. De- termine the probability that the driver is a Bad or Egregious driver. NOTE: Round to the nearest whole percent. Answer in units of percent.arrow_forwardA college radio station surveyed incoming freshmen to gather information about the types of music that they like. Here are the results. • 647 incoming freshmen were surveyed in all • 371 of the freshmen like rap • 82 of the freshmen like jazz, but like neither rap nor classical 47 of the freshmen like rap, but like neither classical nor jazz • 38 of the freshmen like classical, but like neither rap nor jazz • 133 of the freshmen like both rap and classical, but don't like jazz • 123 of the freshmen like both rap and jazz, but don't like classical • 545 of the freshmen like at least one of these three types of music (rap, classical, jazz) Construct a Venn diagram illustrating these results. Then answer the question. All freshmen in the survey How many of the freshmen like classical? freshmen Like rap Like classical Like jazz Continue Submi 2021 McGraw-Hill Education. All Rights Reserved. Terms of Use Privac MacBook Airarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
![Text book image](https://www.bartleby.com/isbn_cover_images/9780079039897/9780079039897_smallCoverImage.jpg)
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
![Text book image](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
![Text book image](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:Pearson Addison Wesley,
What is a Relation? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=hV1_wvsdJCE;License: Standard YouTube License, CC-BY
RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY