Understanding True/False Statements in Logic Fundamentals

4.1 True/false statements True/false statements Logic is the ±eld of precise reasoning. If people under age 7 can't ride, and Jan is under age 7, then Jan can't ride. That's logic in action. At the heart of logic are true/false statements , which are statements that are either true or false, such as "Jan is under age 7." The truth value of a true/false statement indicates whether the statement is true or false based on the data or facts given. Given the fact that Jan was born 8 years ago, the truth value of the statement "Jan is under age 7" is false. Logic builds on such true/false statements and truth values. PARTICIPATION ACTIVITY 4.1.1: True/false statements and truth values. PARTICIPATION ACTIVITY 4.1.2: True/false statements. Indicate whether the statement is a true/false statement. 1) Joe passed Chem101. Yes No 2) Joe can take Chem102. Yes No Animation content: unde±ned Animation captions: 1. Logic starts with statements that are either true or false. Ex: "Lee passed Chem101" must be true or false. If Lee got a B: true. If an F: false. If Lee didn't take the class: false. 2. "Mia is a parent" is a true/false statement, because the statement is either true or false. If Mia has no kids, the statement is false. Later, if Mia has 3 kids, the statement is true. 3. "Good morning" is not a true/false statement, as no truth value exists for such a greeting. "What day is today?" is not a true/false statement either, instead being a question. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

3) Joe is a parent. Yes No 4) Are the kids sleepy? Yes No 5) Hello everybody. Yes No 6) Kay does not have ID. Yes No 7) Kay might win the lottery. Yes No 8) The earth is ²at. Yes No PARTICIPATION ACTIVITY 4.1.3: Truth value of true/false statements. For given facts, a true/false statement's truth value is either true or false. For the facts given below, indicate whether the subsequent true/false statement's truth value is true or false. 1) Fact: Stacy got a B in Chem101. Statement: Stacy passed Chem101. True False 2) Fact: Stacy got an F in Chem101. Statement: Stacy passed Chem101. True False ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

3) Fact: Mia has three children. Statement: Mia is a parent. True False 4) Fact: Mo is 5 years old. Statement: Mo is at least 4 years old. True False 5) Fact: Ladybugs can't bite people. Statement: A ladybug can bite people. True False 6) Fact: Today is Monday. Statement: A week has 7 days. True False Does your dog bite? Just for fun, this classic scene (77 sec) from an early Pink Panther movie shows a misunderstanding around the truth value of a true/false statement "My dog does not bite". The dog part starts at 50 sec. The negation of a true/false statement People apply logic in everyday life. A common logic task is to negate a true/false statement, or to change the statement so that the truth value is reversed. Unfortunately, the nature of English can lead to mistakes. In everyday life, people don't say "Not Mia is a parent" but rather "Mia is not a parent", so negation in English is harder. To avoid mistakes, logicians use letters like P, so the negation of P is "NOT P". Photo source: Screenshot from the above video clip on youtube.com, 2019. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.1.4: The negation of a true/false statement. PARTICIPATION ACTIVITY 4.1.5: Expressing in English that a true/false statement is false. Choose the most appropriate negation of the true/false statement. 1) Joe broke the theft law. Joe did not break the theft law. No person broke the theft law. Joe broke no laws. 2) Cleo is barking. Cleo is quiet. Cleo is not barking. 3) The package has been opened. The package is unopened. The package has not been touched. The package is wide open. PARTICIPATION ACTIVITY 4.1.6: Selecting a good negation of a true/false statement. Indicate whether the negation is good or poor. Animation content: unde±ned Animation captions: 1. "Mia is a parent" is a true/false statement. Logicians use letters for such statements, like P. If P is true, then the negation NOT P is false. 2. Unfortunately, people can't use the same approach for the negation in regular English. Nobody says "Not Mia is a parent", but instead "Mia is not a parent". 3. Such negation in English is prone to errors. The negation of "Jan is under age 7" is "Jan is 7 or older", rather than "Jan is over age 7", because Jan can be 7. 4. The negation of "The food is expensive" should not be "The food is cheap" because the meaning may change (cheap implies bad). A better negation is "The food is inexpensive". ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

1) Statement: The player is under 10 years old. Negation: The player is over 10 years old. Good Poor 2) Statement: Mo is short. Negation: Mo is tall. Good Poor 3) Statement: Jan is a fast worker. Negation: Jan is a slow worker. Good Poor 4) Statement: The babysitter is reliable. Negation: The babysitter is unreliable. Good Poor PARTICIPATION ACTIVITY 4.1.7: Common survey errors involving negating. Logic mistakes commonly appear in surveys. Indicate whether the pair of choices are good or poor, based on whether the second choice is a good negation of the ±rst. 1) Choice 1: I work less than 20 hours per week. Choice 2: I work more than 20 hours per week. Good Poor 2) Choice 1: I am currently employed by the government. Choice 2: I am not currently employed by the government. Good Poor ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

3) Choice 1: The exam was easy. Choice 2: The exam was hard. Good Poor So you think I'm fat? People commonly make logical negation errors in everyday life, especially on subjects to which the person is sensitive. How often has one observed things similar to the following? Do you think I look skinny? No? So you think I'm fat? Am I overdressed? No? So you think I'm underdressed? I asked the manager if Kia was a strong worker and she said no. So maybe we should let Kia go, as we don't want weak workers. News story: "Stocks are not expected to rise." Person talking to a friend: "The news said stocks are expected to fall". That car is not rated as highly reliable. I don't think you should buy that car, because you don't want a car that is unreliable. My son didn't call to wish me happy birthday. He must not care about me. Many everyday problems occur due to incorrectly negating. Garth knows how to negate. Garth from the Saturday Night Live recurring sketches about Wayne's World knows how to negate accurately. Garth just says "I'm having a good time ... NOT", as in this 10-sec video . That works! Too bad using that approach in real life is not socially acceptable. Photo source: Screenshot from above youtube video, 2019. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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CHALLENGE ACTIVITY 4.1.1: True/false statements. 525966.2980244.qx3zqy7 Start 2 3 Indicate whether the statement is a true/false statement: How are you doing? Check Next 4.2 If-then statements If-then statements One form of logic uses an if-then statement : A statement relating two true/false statements, saying that IF statement 1 is true, THEN statement 2 is also true. PARTICIPATION ACTIVITY 4.2.1: An if-then statement relates two true/false statements: If statement1 is true, then statement2 is also true. Animation content: unde±ned Animation captions: 1. An if-then statement relates two true/false statements: If statement 1 is true, then statement 2 is also true. 2. Lee got a B in Chem101. Because "Lee passed Chem101" is true, then "Lee can take Chem102" is also true, according to the if-then statement. 3. When statement 1 is false, the if-then statement says nothing about statement 2's truth value. Maybe Lee can still take Chem102 with instructor permission, by passing a test, etc. 1 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

PARTICIPATION ACTIVITY 4.2.2: If-then statements. Indicate whether "Bandit will bark" (Bandit is a dog) is known to be true, based solely on the info given and this if-then statement: If Bandit is scared, then Bandit will bark. 1) "Bandit is scared" is true. True (Bandit will bark) False (Bandit will not bark) Unknown (whether Bandit will or will not bark) 2) Keesha scared Bandit. True (Bandit will bark) False (Bandit will not bark) Unknown 3) "Bandit is scared" is false. True (Bandit will bark) False (Bandit will not bark) Unknown Confusing terminology can slow learning. The ±eld of logic uses different terms than in this material: A true/false statement is called a proposition . An if-then statement is called a logical implication . But, those words have different meanings in everyday English, where a "proposition" is a suggestion like "I think we should get married", and an "implication" is something that can be concluded indirectly as in "His sloppy appearance implies he doesn't want this job." Just as in the math and statistics ±elds, the logic ±eld's rede±ning of common words can cause confusion and slow the learning of concepts. So, this material uses more intuitive terms. If studying formal logic further, the reader will want to learn the ±eld's more common terms. Everyday if/then statements ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

If-then statements appear in everyday life, even if not always written in if-then form. PARTICIPATION ACTIVITY 4.2.3: If-then statements in everyday life. PARTICIPATION ACTIVITY 4.2.4: Detecting if-then statements from English sentences. Indicate the most appropriate if-then statement that the English sentence represents. 1) If Cleo sees a stranger, Cleo will bark. If Cleo sees a stranger, then Cleo will bark. If Cleo barks, then the Cleo saw a stranger. 2) Travelers not present 45 min before scheduled departure are subject to loss of seat. If a traveler isn't present 45 min before scheduled departure, then the traveler is subject to loss of seat. If a person's seat was lost, then the traveler was not present 45 min before scheduled departure. Animation content: unde±ned Animation captions: 1. If-then statements are common in everyday life. A hotel policy may state: If you cancel by 6 pm, you will not be charged. The "then" before "you will not" is understood. 2. Often statements are written differently, as in: Cancellations made by 6 pm will not be charged. Or perhaps: You will not be charged if you cancel by 6 pm. 3. A country may state this if-then statement in various ways: If you don't have a passport, you will be denied entry. The ability to mentally derive the if-then statement logic is useful. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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3) A package cannot be returned if the package has been opened. If a package cannot be returned, then the package was opened. If a package has been opened, then the package cannot be returned. If a package has not been opened, then the package can be returned. 4) A free kick is awarded when a player trips an opponent. If a player trips an opponent, then a free kick is awarded. If a free kick is awarded, then a player tripped an opponent. CHALLENGE ACTIVITY 4.2.1: If-then statements. 525966.2980244.qx3zqy7 Start 2 3 4 Based on the following correct if-then statement and given fact, select whether the last statement is true, false, or unknown. If the weather is rainy, then the garden does not need to be watered. Fact: The weather is not rainy. Statement: The garden does not need to be watered. Check Next 4.3 Logical deduction 1 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

Logical deduction Perhaps the best-known form of logic is logical deduction , or using one statement's truth value to determine another statement's truth value. Given a correct if-then statement, if statement 1 is true, then a person can deduce that statement 2 is true. Interestingly, if statement 2 is false, then a person can also deduce that statement 1 is false. PARTICIPATION ACTIVITY 4.3.1: If-then statements and logical deduction. PARTICIPATION ACTIVITY 4.3.2: Basic deduction from an if-then statement. Given the if-then statement: If A, then B. What can be deduced? 1) A is true, so B is true. Yes No 2) A is false, so B is false. Yes No Animation content: unde±ned Animation captions: 1. Logical deduction involves determining one true/false statement's truth value from another statement's true value. If A is true, then B is true. 2. Given the statement: If Pat just drove the car, then Pat's engine is hot. Given data that Pat just drove the car is true, then Pat's engine is hot is also true. That's logical deduction. 3. But, if "Pat just drove the car" is false, nothing is known about the engine being hot. Maybe somebody else drove heated the engine, or not so the engine is cold. 4. Interestingly, if the engine is NOT hot, then Pat certainly did NOT just drive the car. (If Pat had just driven, the engine would be hot). That's also logical deduction. 5. In other words, given "If A, then B", if data shows B is false, then A is also false. ³. But, given that the engine is hot, nothing is known about whether Pat just drove the car. Somebody clearly drove, but who is unknown. 7. In summary, given "If A, then B", A is true means B is true, and B is false means A is false. Both are logical deductions. For the other two cases, nothing can be deduced. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

3) B is false, so A is false. Yes No 4) B is true, so A is true. Yes No PARTICIPATION ACTIVITY 4.3.3: Did Pat just drive the car? A police o´cer investigates a hit-and-run. Somebody says they saw Pat's car leaving the hit- and-run scene, so the o´cer goes to Pat's home, just 10 minutes after the hit-and-run. Consider the correct if-then statement: If Pat was just driving, then Pat's engine is hot. 1) Pat says "I've been home all day. Go feel my car's engine; if the engine is cool, you know I wasn't just driving the car." Is Pat's logic correct? Yes No 2) The o´cer feels Pat's car engine, which is hot. The o´cer says "The engine is hot, so you were just driving." Is the o´cer's logic correct? Yes No PARTICIPATION ACTIVITY 4.3.4: Uses of logical deduction. Which are proper applications of logical deduction? 1) Everyone agrees that if Val robbed a house last night, then Val must have been in that house last night. To prove Val did not rob the house, Val's defense attorney seeks to prove Val was not in the house last night. OK Not OK ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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2) Everyone agrees that if Val robbed a house last night, then Val must have been in that house last night. To prove Val robbed the house, a prosecutor seeks only to prove Val was in the house last night. OK Not OK Chained deductions People commonly make deductions by chaining several if-then statements. PARTICIPATION ACTIVITY 4.3.5: Chained deductions. PARTICIPATION ACTIVITY 4.3.6: Chained deductions. Consider the example above, which includes the if-then statements "If A, then B" and "If B, then C". 1) If A is true, can one deduce that C is true? Yes No Animation content: unde±ned Animation captions: 1. Given "If A, then B" plus "If B, then C", and fact "A is true". Then B must be true. Since B is true, then C must be true. People commonly make such chained deductions. 2. Or, given that C is false, one can deduce that B is false, and then deduce that A is false. 3. If Jo has a license, then Jo is over 15, and so then Jo can enter. Checking Jo's date of birth on the license is not necessary. 4. Given that Jo was too young to enter, then Jo is not over 15, so then Jo cannot have a driver's license. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

2) If A is false, can one deduce that C is false? Yes No 3) If B is false, can one deduce that A is false? Yes No 4) If B is false, can one deduce that C is false? Yes No 5) If C is true, can one deduce that A is true? Yes No Witches and chained deductions. This classic Monty Python movie scene (2 min) shows particularly erroneous chained logical deductions. CHALLENGE ACTIVITY 4.3.1: Logical deduction. 525966.2980244.qx3zqy7 Photo from video clip of the movie "Monty Python and the Holy Grail", taken from above link. Start Given: If A, then B. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

2 Indicate what is known for the given data: B is true. A is Check Next 4.4 If-then statements whose reverse is correct If-then statements whose reverse is correct too Sometimes an if-then statement works not just forwards but also in reverse, meaning that both "If A, then B" and "If B, then A" are correct. An if-then statement whose reverse is also correct allows a person to deduce more than just a one-way if-then statement. PARTICIPATION ACTIVITY 4.4.1: If-then statements whose reverse is correct. Note: Logicians indicate if-then statements whose reverse is also correct by saying "A if and only if B", or "A iff B" for short. Logicians write a regular if-then statement using the notation A → B, and an if-then statement whose reverse is correct as A ↔ B. Animation content: unde±ned Animation captions: 1. An if-then statement only allows deduction when A is true, not false. When A is true, then B must be true. When A is false, the statement says nothing about B. 2. Given: If Bandit is scared, Bandit will bark. But when Bandit isn't scared, nothing can be deduced about barking. Bandit might bark anyway if angry or excited. 3. Also, if B is known to be false, then A can be deduced to be false. But, B being true doesn't mean A is true. 4. However, for some if-then statements, the reverse is also correct. Correct: If today is Mon, then tomorrow is Tue. Also correct: If tomorrow is Tue, then today is Mon. 5. For such a case, B can ONLY be true if A is true. If A is false, B must be false. And, if B is true, A must be true. 1 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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In everyday English, sometimes people will say something like "If A, then B, and vice-versa", as in "Those twins are something. If Jo cries, then Kay cries, and vice-versa." However, in everyday English, people often just state the forward if-then, with the reverse being understood using common sense. That situation can unfortunately cause confusion, but English isn't perfect. Logic requires focus A note to readers: Understanding logic requires a lot of mental focus. If the reader has been enjoying music (or maybe even YouTube videos) while learning material in earlier chapters, the reader is encouraged to turn the music off for a while, and really let the brain focus on the logic. The reader may ±nd that exercise challenging and fun. PARTICIPATION ACTIVITY 4.4.2: Determining the truth value of a forward conditional statement and the converse. Use everyday knowledge to determine whether the conditional statement is only true forward, or if the converse is also true too. 1) If rain is falling, then the sky has clouds. Forward only Forward and converse 2) If today is Saturday, then today is the weekend. Forward only Forward and converse ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

3) If smoke exists near a smoke detector, then a functioning smoke detector will sound. Forward only Forward and converse 4) If a lamp is unplugged, then the lamp will not light. Forward only Forward and converse 5) If a car was recently running, then the engine will be hot. Forward only Forward and converse 6) A de±nition of shoplifting is a person intentionally removing an item from a store without paying. Stated as a conditional statement: If a person intentionally removes an item from a store without paying, that person has committed shoplifting. Forward only Forward and converse A note to readers: For nearly any example above, some readers will ±nd an exception. Maybe an unplugged lamp will light via an internal battery. Maybe a smoke detector will sound due to malfunctioning. Maybe once in history the day after Monday wasn't Tuesday because the government adjusted the calendar ( that really happened ). The reader is asked, for this section and others, to assume the common sense situation applies. Otherwise, logic has to be taught mostly using symbols like A and B, but this material strives to teach logic in context of everyday life. Logical deduction with if-then statements whose reverse is correct As seen in the animation above, an if-then statement whose reverse is correct allows for more deductions. In a if-then statement "If A, then B" whose reverse is correct, both A and B must be true, or both must be false. So knowing the truth value of either A or of B, a person can deduce the other. PARTICIPATION ACTIVITY 4.4.3: Deducing from if-then statements when given that the reverse is correct. Given: For each if-then statement below, the reverse is also correct. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

1) If A, then B. A is true. So B is ___. true false 2) If A, then B. A is false. So B is ___. true false 3) If A, then B. B is true. So A is ___. true false 4) If A, then B. B is false. So A is ___. true false 5) If a baby needs something, a baby cries. The baby is crying. So the baby ___ something. needs may need CHALLENGE ACTIVITY 4.4.1: If-then statements whose reverse is correct. 525966.2980244.qx3zqy7 Start 2 3 4 5 Using everyday knowledge, indicate whether the if-then statement's reverse is also correct. X: If rain is falling, then the sky has a cloud. Y: If Jo is taller than Lee, then Lee is shorter than Jo. X is Y is 1 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Check Next A correct if-then statement is: If today is Monday, then tomorrow is Tuesday. An incorrect if-then statement is: If today is Monday, then tomorrow is Friday. 4.5 Common logical deduction mistakes Incorrect if-then statements A common logic-related mistake is to start from an incorrect if-then statement. An if-then statement like "If A, then B" might be said to be correct if A being true indeed means B is also true. Most incorrect cases aren't as obvious. "If an employee is often late to work, then the employee must be irresponsible." But, maybe the employee is visiting an ill parent in the mornings, and working late to make up the time? Interestingly, this error of attributing behavior to character ²aws instead of external situations is so common as to be known as the fundamental attribution error . "If a poorly-dressed person is lying on the sidewalk, then the person is homeless." Many people make that assumption, which is often correct. But maybe the person was heading to work, fell ill, and needs help? This video (3-min) shows an experiment where people didn't help a person lying on the sidewalk, except when the person was dressed well. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

PARTICIPATION ACTIVITY 4.5.1: Correct and incorrect if-then statements. Indicate which if-then statements are most likely to be correct or incorrect. 1) If a car won't start, something is wrong with the car. Correct Incorrect 2) If a new coworker doesn't smile when introduced, the coworker is unfriendly. Correct Incorrect 3) If I don't get a call from my adult child on my birthday, then my child no longer cares about me. Correct Incorrect 4) If my neighbor says my haircut looks great, then my neighbor thinks my hair looked bad before. Correct Incorrect Photo taken from above video link on youtube, 2019 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

If Mia has at least one kid, then Mia is a parent / If Mia is a parent, then Mia has at least one kid If today is Monday, then tomorrow is Tuesday / If tomorrow is Tuesday, then today is Monday 5) If my spouse isn't saying much during this car drive, then my spouse must be upset with me. Correct Incorrect 6) If Jo babysits, Pat will pay Jo. Given this info: Jo did babysit but Pat did not pay Jo.. Correct Incorrect Note that an if-then statement is itself a true/false statement. So an incorrect if-then statement is like a true/false statement with a truth value of false. This concept can be a little hard to get one's head around, so this material won't discuss that point further. Common logic mistake: Assuming if-then applies in both directions In everyday life, many if-then statements have a reverse that is also correct, as in the following: Because if-then statements whose reverse is correct are so common, many people mistakenly treat all if- then statements as having correct reverses too, leading to incorrect deductions. If a person has a contagious cold/²u, then the person will cough / If a person is coughing, then the person has a contagious cold/²u. But the second true/false statement may be incorrect; the person may have just eaten something that irritated the throat. If a person is drunk, then the person will wobble when walking / If a person wobbles when walking, then the person is drunk. But the second true/false statement may be incorrect; the person may have a disability, as airline employees and others often ±nd out the hard way when seeing someone who wobbles when walking or has slurred speech. Screenshot source: Headline from above news article link, 2017. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.5.2: If-then statements with and without correct reverses. Indicate which everyday if-then statements are only correct forward (one-way) or whose reverse is also correct (both ways). 1) If Jo is Mia's child, then Mia is Jo's parent. One-way Both ways 2) If Pat is 16 now, then Pat was 15 a year ago. One-way Both ways 3) If a person has trouble walking, then the person is (at least momentarily) disabled. One-way Both ways 4) If a person has a contagious rash, then the skin will show the rash. One-way Both ways 5) If my spouse is upset with me, my spouse won't speak during a car drive. One-way Both ways A common if-then statement both-way assumption after a disaster. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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A particularly sad both-way assumption error is often heard after a natural disaster. After a hurricane devastates a city, one might hear "Well, those people chose to live in a hurricane zone." The same is heard for ±res, tornadoes, ²oods, and other disasters. Many people believe this if-then statement to be correct: If people knowingly choose to live in a very dangerous area, then those people are likely to experience devastation. But after a disaster, many people apply that if-then statement in reverse: If people experienced devastation, then those people knowingly chose to live in a dangerous area. But, just because devastation occurs in an area doesn't mean the area was known to be dangerous. Even the east coast of Florida, which gets hit by many hurricanes, isn't necessarily a dangerous area; the vast majority of homes experience no problems. That logic error is likely a way of avoiding the pain a person experiences when empathizing with others' suffering. But sometimes feeling that pain is perhaps what's the most appropriate and human thing to do. Photo sources: (Left) nypost.com , (Right) time.com 4.6 Logic: AND/OR True/false statements with AND/OR A true/false statement may combine two (or more) other true/false statements, connected by AND or OR. The logical AND function evaluates to true only if all sub-statements are true. The logical OR function evaluates to true if any sub-statement is true. PARTICIPATION ACTIVITY 4.6.1: AND and OR logic functions. Animation content: unde±ned ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.6.2: True/false statements involving AND/OR. Below, A as well as B are true/false sub-statements. 1) What is A AND B, when A is false and B is true? True False 2) What is A AND B, when A is true and B is true? True False 3) What is A OR B, when A is false and B is true? True False 4) What is A OR B, when A is true and B is true? True False 5) What is A OR B, when A is false and B is false? True False Animation captions: 1. A person being at least 36 inches tall is a true/false statement, as is being at least 4 years old. The statements can be combined using AND, forming a new true/false statement. 2. A table can be used to show the truth value of the AND statement depending on the truth values of the two sub-statements. 3. AND evaluates to true only when both sub-statements are true. If a person is 38 inches tall and 5 years old, both sub-statements are true, so the person can ride. 4. But if a person is only 32 inches tall, the ±rst sub-statement is false, so the person cannot ride (no matter whether the second sub-statement is true). 5. An OR statement is true if only one or both sub-statements are true. If either a person is under 30 inches OR under 7 years, or both, a car seat is required. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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6) For the example above, can a 40-inch- tall 4-year-old ride? Yes No 7) For the example above, is a car seat required for a 60-inch-tall 6-year-old? Yes No 8) AND can apply to more than two sub- statements. To what does A AND B AND C evaluate when A is true, B is true, C is false? True False 9) OR can apply to more than two sub- statements. To what does A OR B OR C evaluate when A is true, B is false, C is false? True False 10) As with arithmetic expressions like (4 + 2) * 3, logic expressions can be longer and involve parentheses. To what does (A AND B) OR C evaluate when A is true, B is true, C is false? True False PARTICIPATION ACTIVITY 4.6.3: AND/OR logic in everyday English. In everyday English, the words AND or OR are sometimes omitted. People must use common sense to recognize whether the English meaning is AND or OR. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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1) Does the following use AND logic or instead use OR logic? To enroll in the ABC preschool, students must: (1) Be 4 years old or older, (2) Have up-to-date immunizations. AND OR 2) Does the following use AND logic or instead use OR logic? These jobs qualify for a museum discount: (1) Teacher, (2) First- responder, (3) Daycare worker. AND OR OR in everyday English is sometimes exclusive In everyday English, sometimes OR is used as above. For a rated-R movie, children under 17 can enter if accompanied by a parent or guardian. A parent, guardian, or both, enables a child to enter. But sometimes, OR is used in an exclusive manner. A menu says that a meal comes with salad or soup. The OR clearly means only salad or only soup, but not both. In logic, this form of OR is called "exclusive OR". Unfortunately, English uses the same word for both concepts (aargh!). The reader should realize that the ±eld of logic, and this material, uses OR to refer to the "inclusive" form of OR (the parent/guardian example above), and will say "exclusive OR" explicitly if using that form. Negating an AND or OR expression Some simple math increases the power of logic. A common example is ±nding the negation of an expression involving AND and OR, where the new expression evaluates to the opposite of the original. The rules for negating an AND/OR expression are known as De Morgan's Laws . De Morgan's Laws ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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The negation of A AND B, or NOT(A AND B), is NOT(A) OR NOT(B). The negation of A OR B, or NOT(A OR B), is NOT(A) AND NOT(B). PARTICIPATION ACTIVITY 4.6.4: Negating an AND or OR expression. PARTICIPATION ACTIVITY 4.6.5: Negating a logic expression with AND / OR. Indicate if the negation was done correctly. 1) NOT(A AND B) = NOT(A) OR B Correct Incorrect 2) NOT(A AND B) = NOT(A) AND NOT(B) Correct Incorrect Animation content: unde±ned Animation captions: 1. In English, negating a logical AND expression can be tricky. What is the negation of a rule requiring a rider to have a wristband AND that the rider is at least 5 years old? 2. De Morgan's Law helps. NOT(A AND B) = NOT(A) OR NOT(B). Each term is negated, and the AND changes to OR. 3. Negating the English statement yields: NOT(rider has wristband) OR NOT(rider at least 5 yrs). 4. Negating each sentence yields: Rider lacks wristband OR rider under 5 yrs. The rider cannot ride if either is true. 5. Negating an expression with OR is similar. In that case, the OR becomes an AND. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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3) NOT(A AND B) = NOT(A) OR NOT(B) Correct Incorrect 4) NOT(A OR B) = NOT(A) OR NOT(B) Correct Incorrect 5) NOT(NOT(A)) = A Correct Incorrect 6) NOT(NOT(A) AND B) = NOT(NOT(A)) OR NOT(B) = A OR NOT(B) Correct Incorrect If-then statements with AND/OR logic In an if-then statement, the ±rst statement is often composed of sub-statements connected by AND or OR. PARTICIPATION ACTIVITY 4.6.6: If-then statements with AND/OR. PARTICIPATION ACTIVITY 4.6.7: If-then statements involving AND or OR logic. Animation content: unde±ned Animation captions: 1. The IF part of an if-then statement is commonly built from sub-statements connected by AND or OR. 2. If either sub-statement in the IF part is true, then "Must use car seat" is true. 3. Such if-then statements are commonly written differently in English, but can be converted to if- then statement form. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Below, A, B, C are each a true/false statement. Assume any given if-then statement is correct, and only applies one way (as with any if-else statement unless otherwise noted or understood). 1) If A AND B, then C. What is C when A is true and B is true? True False Unknown 2) If A AND B, then C. When A is true and B is false, what is C? True False Unknown 3) If A OR B, then C. When A is true and B is false, what is C? True False Unknown 4) If a dog is scared or angry, the dog will bark. Fact: The dog is angry. Will the dog bark? Yes No Unknown 5) People must be at least 18 and in good health to enter. Fact: Joe is 20. Joe is in good health. Can Joe enter? Yes No Unknown ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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6) If a dog is scared or angry, the dog will bark. Fact: The dog is not scared. The dog is not angry. Will the dog bark? Yes No Unknown CHALLENGE ACTIVITY 4.6.1: Logic: AND/OR/NOT. 525966.2980244.qx3zqy7 Start 2 3 4 5 Given: A is true, B is true. A AND B is A OR B is Check Next 4.7 AND/OR/NOT logic examples Example: Web searches Web searches on google.com and other web search engines involve AND and OR logic. The words in a search by default are treated as being joined by AND. Ex: A search for "²u cold" is interpreted as "²u AND cold", meaning found pages should discuss both ²us and colds. In fact, a person can type that AND between words without changing the meaning of the search. In contrast, typing "²u OR cold" means found pages should discuss either ²us or colds (perhaps both). Search tools consider other factors when returning search results, but search strongly favors pages that satisfy AND/OR logic. 1 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Figure 4.7.1: A search for ²u cold yields pages that each discuss both, but a search for ²u OR cold yields 5 pages where 2 discuss only the ²u. The same logic is used for email searches, photo searches, and more. A photo search for "cats dogs" will favor photos having both a cat AND a dog in the photo, while "cats OR dogs" will favor photos having either a cat, or a dog, or both. Figure 4.7.2: Left: A search for cat dog in this person's photos yields no results. Right: A search for cat OR dog yields numerous results, of just dogs. Clearly this person is a dog-lover, and apparently doesn't have much interest in cats. Image source: Google search results screenshots, 2019. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.7.1: Web searching and AND/OR logic. What pages will be favored for the given search? 1) Julia AND Roberts Pages about Julia Roberts Pages with the words Julia, AND, and Roberts Pages with either the word Julia, the word Roberts, or both 2) New York Pages about New York Pages about New, or about York, or about both. 3) Bees OR wasps Pages that include discussion of both bees and wasps on the same page. Pages that discuss bees only, or that discuss wasps only, or that discuss both. Pages that contain the words bees, OR, and wasps on the same page. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.7.2: Logic with AND and OR. On a jet with two bathrooms, when bathroom 1 is occupied AND bathroom 2 is occupied, an "all-bathrooms-are-occupied" red light turns on. In other words: IF bathroom 1 is occupied AND bathroom 2 is occupied, THEN the red light turns on. The if-then statement's reverse is also correct: IF the red light is on, THEN bathroom 1 is occupied AND bathroom 2 is occupied. 1) Fact: Bathroom 1 is occupied. Bathroom 2 is occupied. Light is off Light is on 2) Bathroom 1 is occupied, but Bathroom 2 is not occupied. Light is off Light is on 3) The light is red. Both bathrooms are occupied At least one bathroom is unoccupied ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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4) Some signs don't just turn from on to off when a bathroom is available, but instead turn the light green. A person sees the following bathroom-occupied sign lit green. Both bathrooms are unoccupied At least one bathroom is unoccupied Example: Jaywalking laws Laws make extensive use of AND/OR logic. For example, California law de±nes jaywalking as a person crossing a street outside a crosswalk where the intersection to the left is controlled by a signal AND the intersection to the right is controlled by a signal. (A signal is a tra´c light, meaning those red, yellow, and green lights people see regularly). The law appears here . PARTICIPATION ACTIVITY 4.7.3: California jaywalking law involves AND logic. Laws are usually if-then statements whose reverse is also correct. If speci±c things are done, then a person broke that law. AND, if the person broke that law, then the person did those speci±c things. In other words, the ONLY way to jaywalk is to cross AND the left intersection is a signal AND the right intersection is a signal. PARTICIPATION ACTIVITY 4.7.4: Jaywalking in California. Animation content: unde±ned Animation captions: 1. CVC 21955: Between adjacent [signal-controlled] intersections, pedestrians shall not cross the roadway [unless in a crosswalk]. 2. As an IF-THEN statement: IF a person crosses AND left intersection is signal AND right intersection is signal, THEN jaywalking has occurred. 3. If the left has a signal but the right has a stop sign, jaywalking did not occur. All three sub- statements must be true: Crossed AND left is signal AND right is signal. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Indicate whether jaywalking has occurred if a person crosses the street (outside a crosswalk), between left and right intersections as noted. 1) Left: Signal Right: Signal Yes No 2) Left: Stop sign Right: Signal Yes No 3) Left: Stop sign Right: Stop sign Yes No 4) The jaywalking law is an if-then statement whose reverse is also correct. A person is guilty of jaywalking. Can one deduce that the person crossed AND left was signal AND right was signal? Yes No 5) The actual law just says "controlled" rather than "signal-controlled", and de±nes an intersection as controlled if the intersection has a signal OR an o´cer. Does this sentence describe the law's logic? Jaywalking occurs if: A person crosses outside a crosswalk AND (left is signal OR o´cer) AND (right is signal OR o´cer) Yes No True story: Jaywalking ticket ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Tim was a student at a University of California college campus. He was walking home from studying and safely crossed a street without any cars in sight, at a location between a stop sign and a signal. He was stopped by a campus police o³cer who gave him a jaywalking ticket. The o³cer believed the law meant that either intersection could be a signal. Tim read the jaywalking law, and argued to a judge in court that the o³cer misunderstood the logic and in fact both intersections had to be signals. Tim's ticket was overturned by the judge. Example: Car seat usage The California law for car seat usage appears below. The law involves a lot of AND and OR logic. Figure 4.7.3: California car seat law (2019). Children under 2 years of age shall ride in a rear-facing car seat unless the child weighs 40 or more pounds OR is 40 or more inches tall. Children under the age of 8 must be secured in a car seat or booster seat in the back seat. Children who are 8 years of age OR have reached 4'9" in height may be secured by a booster seat, but at a minimum must be secured by a safety belt. Below are the same rules, rewritten using if-then statements. Figure 4.7.4: California car seat law written as if-then statements. IF (child age is under 2 years) AND NOT( (child weight is 40+ pounds) OR (child height is 40+ inches) ), THEN must ride in rear-facing car seat. IF (child age is under 8 years), THEN must ride in back seat in ( car-seat OR booster-seat ). IF ( (child age is 8+ years) OR (child height is 4'9"+), THEN must use (booster-seat OR safety-belt). PARTICIPATION ACTIVITY 4.7.5: Car seat law. Consider the car seat law above. Source: California Highway Patrol ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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1) May a 1.5 year old child weighing 30 pounds and with height 30 inches ride in a forward-facing car seat? Yes No 2) May a 5 year old weighing 70 pounds ride in the front seat? Yes No 3) May a 9 year old weighing 38 pounds use just a seat belt? Yes No 4.8 Common applications of logic Laws Laws make extensive use of if-then logic (usually both-way: the reverse is also correct)), including AND/OR logic. In everyday life, people need to understand laws to avoid breaking laws. Plus, many people may serve on a jury someday, where judges ask jury members to evaluate the logic of a law carefully. Juries only decide what are the facts of a case, namely which facts are true or false. Given those facts, juries should apply the law's logic, whether a juror agrees with the logic or not. Figure 4.8.1: Laws on calling police after an accident. Many people don't realize that most states have a law that requires a person to contact the police after a car accident, sometimes immediately, sometimes within a few days. Below are a few such laws from various states, taken from AAA.com (2019). Many such laws make use of OR logic: IF injury OR death OR damage exceeds some amount, THEN police must be noti±ed. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Arizona: Crashes are required to be reported when property damage exceeds $300.00, results in bodily injury to or death of a person, or if the settlement of the crash has not been reached within 6 months of the crash. California: Crashes are required to be reported in cases of death or injury or when property damage exceeds $1,000. Reports must be ±led within 10 days. Florida: Crashes are required to be reported as soon as possible in cases of death or injury, or when property damage exceeds $500.00. Nevada: Crashes are required to be immediately reported. Washington: Crashes are required to be reported in cases of death, injury, or when property damage exceeds $700.00. The deadline to ±le a report is 4 days. PARTICIPATION ACTIVITY 4.8.1: Contacting police after an accident. Jay was involved in a car accident yesterday. Damage is about $500, no injuries, no deaths. According to the laws listed above, is Jay required to contact the police in the given state? Note: Even if not required, many people recommend calling the police anyways to have them come to the accident scene and ±le a report. 1) California Yes No 2) Arizona Yes No 3) Nevada Yes No Ouch. Frank's daughter was in another accident. Everyone was OK. Phew. This time the truck survived too. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Figure 4.8.2: Laws on leaving animals in cars. A car's inside can heat up quickly when parked in the sun. 80°F outside can turn into well over 100°F inside. Thus, many states now have laws against endangering animals in parked cars. Furthermore, many states have laws granting immunity to people who break into a car to save an animal in danger. Below are those two laws in California Penal Code 597.7 : A person shall not leave or con±ne an animal in any unattended motor vehicle under conditions that endanger the health or well-being of an animal due to heat, cold, lack of adequate ventilation, or lack of food or water, or other circumstances that could reasonably be expected to cause suffering, disability, or death to the animal. A person who removes an animal from a vehicle [believing the animal to be in immediate danger] is not criminally liable ... if the person does all of the following: (A) Determines the vehicle is locked or there is otherwise no reasonable manner for the animal to be removed from the vehicle. (B) Has a good faith belief that forcible entry into the vehicle is necessary because the animal is in imminent danger of suffering harm if it is not immediately removed from the vehicle, and, based upon the circumstances known to the person at the time, the belief is a reasonable one. (C) Has contacted a local law enforcement agency, the ±re department, animal control, or the "911" emergency service prior to forcibly entering the vehicle. (D) Remains with the animal in a safe location, out of the elements but reasonably close to the vehicle, until a peace o´cer, humane o´cer, animal control o´cer, or another emergency responder arrives. (E) Used no more force to enter the vehicle and remove the animal from the vehicle than was necessary under the circumstances. (F) Immediately turns the animal over to a representative from law enforcement, animal control, or another emergency responder who responds to the scene. PARTICIPATION ACTIVITY 4.8.2: Rescuing animals from hot vehicles. Consider the California laws above regarding leaving animals in a car. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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1) Kim rescues a dog from a car in 102°F weather by breaking a window, but did not check if the car was unlocked ±rst. Does the law grant Kim immunity? Yes No 2) Riley sees a cat in a car in 102°F weather, and the cat appears to be severely dehydrated. Riley ±nds that every door is locked and no windows are open. Riley breaks a window to rescue the cat, then gives the cat water, then calls 911, then waits in an air- conditioned car, and then turns the cat over to arriving police. Does the law grant Riley immunity? Yes No Soccer rules Soccer is a popular sport among kids worldwide and increasingly in the U.S. too. Many parents must learn the rules (called laws) to serve as coaches or referees, or just to better enjoy the game. Below are a few common soccer laws. Table 4.8.1: Common soccer laws. Law Written emphasizing the logic At the moment of delivering the ball, the thrower must: stand facing the ±eld of play have part of each foot on the touchline or on the ground outside the touchline throw the ball with both hands from behind and over the head from the point where it left the ±eld of play. At moment of delivering ball, IF face-±eld AND (feet on touchline OR feet on ground outside touchline) AND ((behind head AND over head) AND where it left ±eld) THEN throw-in is legal. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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A player is cautioned [yellow card] if guilty of: delaying the restart of play dissent by word or action entering, re-entering or deliberately leaving the ±eld of play without the referee's permission failing to respect the required distance when play is restarted with a corner kick, free kick or throw-in persistent offences (no speci±c number or pattern of offences constitutes 'persistent') IF a player delays restart OR (dissents by word OR dissents by action) OR ((enters OR re-enters OR leaves) AND NOT(had permission)) OR ((corner kick OR free kick OR throw in) AND fails to respect distance) OR commits persistent offences THEN the player is cautioned [yellow card] (Originally-worded law for penalty kick omitted here; appears in IF-THEN form to the right). IF (Ball stationary AND ball on penalty mark) AND Kicker identi±ed AND Goalkeeper is (on goalline AND facing kicker AND between posts AND NOT(touching post OR touching bar OR touching net)) AND Other players are (at least 10 yards away AND behind penalty mark AND on ±eld AND outside penalty area) THEN ready for penalty kick. PARTICIPATION ACTIVITY 4.8.3: Logic and soccer rules. 1) A soccer player is getting ready to throw the ball back onto the ±eld. The player jumps in the air and throws the ball from behind and over the player's head. Is this play allowed? Yes No ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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2) A player abruptly leaves the ±eld without informing the referee, but does not delay the restart of play. Should the referee give the player a yellow card? Yes No 3) The teams are preparing for a penalty kick. This situation means the requirement for the ball to be stationary and on the penalty mark must be met. Yes No 4.9 Logical induction Logical induction Logical deduction yields a certain result. Given a correct if-then statement like "If Fido is scared, then Fido will bark", when data/facts indicate "Fido is scared" is true, then for certain "Fido will bark" will be true too. In contrast, logical induction is a kind of logic that is uncertain: Given that several true/false statements are true, a person generalizes that another statement is true. Ex: If a person sees a baby that cries a lot, then another baby that cries a lot, then a third baby that cries a lot, the person may generalize: Babies cry a lot. PARTICIPATION ACTIVITY 4.9.1: Logical deduction is certain, while logical induction is an informed guess. Animation content: unde±ned Animation captions: 1. In logical deduction, given if-then statements, one true/false statement's truth value can be used to deduce another statement's truth value. If F is true, then A is true. 2. If A is true, then B is true. So if F is true, then B certainly must be true. 3. Another form of logic is induction, which generalizes. Mia loves puppies. Jan loves puppies. Generalizing: "People love puppies". 4. Induction is an informed guess. Perhaps all people really do love puppies. Or perhaps some people don't. While not certain, induction is common and useful. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.9.2: Logical induction. Given the true/false statements are true (facts), select the best logical induction. 1) Val likes apples. Cindy likes apples. Stu likes apples. People like apples People don't like oranges 2) Lee likes apples. Lee like oranges. Lee likes grapes. Lee likes melon. Lee likes apples, oranges, grapes, and melon Lee likes fruit 3) A gallon of milk is heavy. A gallon of orange juice is heavy. A gallon of apple juice is heavy. A gallon of cereal is light. A gallon of anything is heavy A gallon of liquid is heavy 4) Fido heard sirens 10 times today and barked 9 of those times. Fido barks when hearing sirens Nothing should be induced PARTICIPATION ACTIVITY 4.9.3: Logical induction vs. deduction. Indicate whether the logic is induction or deduction. 1) Given: If A, then B. Fact: A is true. So B must be true. Induction Deduction ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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2) Hotel XYZ's rate was high 2 months ago. Hotel XYZ's rate was high last month. Hotel XYZ's rate is high this month. So hotel XYZ is expensive. Induction Deduction Common induction mistakes Induction is a widely used form of logic. In fact, generalizing is largely how humans learn and is essential to human survival. Ex: A child pulls a dog's tail, and the dog angrily snaps, scaring the child. The child generalizes that pulling a dog's tail is dangerous. While essential to survival, logical induction is often wrong. Humans may wish to be aware that many of their beliefs are statements thought to be true, but in fact not always true or just plain false, due to incorrect logical induction. Some logical induction leading to incorrect beliefs about humans is called stereotyping . PARTICIPATION ACTIVITY 4.9.4: Logical induction is important, but is sometimes wrong. Indicate whether the proposition obtained via logical induction is likely to be true. 1) A manager hired two students from University of Spring±eld who turned out to be great workers. The manager decides: Students from University of Spring±eld are great. Likely Unclear 2) A manager hired 50 students from University of Spring±eld and 50 from University of Shelbyville. 45 of the Spring±eld students had to be ±red, while 48 of the Shelbyville students worked out great. The manager decides: Students from University of Shelbyville are better than students from University of Spring±eld. Likely Unclear ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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3) Jay has met a few homeschooling families and ±nds the kids socially awkward. Jay concludes: Homeschooling causes social awkwardness. Likely Unclear 4) Flo notices that one particular dog breed seems to show up a lot in "lost dog" ²yers. Flo concludes: Dogs of that breed are more likely to get loose. Likely Unclear 5) The U.S. Surgeon General states: Smoking kills. In other words: If a person smokes, a person is more likely to die earlier than otherwise. Stan says, "I don't believe that. My Grandpa smoked his whole life and lived to be 92." Stan induces: Grandpa smoked and didn't die early, so smokers don't die early. Is Stan's induction likely to be true? Yes No 4.10 Sets and Venn diagrams Venn diagrams Logic operations like AND and OR are often performed on sets. A set is a collection of items, like all people who have student loans, or all people who have credit-card debt. Performing an AND operation on two sets is called intersection . Ex: The intersection of the set of college students and the set of students above age 19 is the set of college students above age 19. Performing an OR operation on two sets is called union . Ex: The union of the set of college students and the set of college employees is the set of college students, college employees, and college student employees. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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A Venn diagram depicts sets graphically, which can be especially helpful for showing intersection and union. PARTICIPATION ACTIVITY 4.10.1: Venn diagrams. PARTICIPATION ACTIVITY 4.10.2: Venn diagram of people with student loans and with credit card debt. Consider the Venn diagram at the bottom left of the animation above. 1) The XYZ organization wants to help people get out of debt. The organization wants to focus on people who have both student loans and credit card debt, as such people may be in the most trouble. Looking at the Venn diagram above, who should XYZ reach out to? Jo, Kyle Mia, Jo, Kyle, Lee, Sam 2) The ABC organization wants to help everyone who has any kind of student loan or credit card debt. Looking at the Venn diagram above, who should ABC reach out to? Mia, Lee, Sam Mia, Jo, Kyle, Lee, Sam Animation content: unde±ned Animation captions: 1. A set is a collection of items, such as people with student loans. Ex: Jo, Kyle, and Mia have student loans and thus form a set. 2. Another set is people with credit card debt. Ex: Sam, Lee, Kyle, and Jo all have credit card debt. 3. In a Venn diagram, people with loans may be in a blue circle and people with credit card debt in an orange circle. Each person is shown only once, so the circles overlap. 4. The people who have student loans AND credit card debt (intersection) are the people in the overlapping region of the two circles, in this case Jo and Kyle. 5. The people who have student loans OR credit card debt (union) are the people in either or both circles, so Mia, Jo, Kyle, Lee, and Sam. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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3) An organization has the following rule: IF a person has a student loan AND a person has credit card debt, THEN the person quali±es for assistance. In the Venn diagram above, who quali±es? Jo, Kyle Mia, Jo, Kyle, Lee, Sam PARTICIPATION ACTIVITY 4.10.3: Sets and Venn diagrams. Consider the Venn diagram below showing the set of people who have worked over 20 years, and the set of people aged over 50 years, at a particular company. 1) Who is everyone that has worked over 20 years? Mike, Kee Mike, Kee, Stu Mike, Kee, Stu, Flo, Vu, Sal 2) Who is everyone that is aged over 50 years? Mike, Kee, Stu, Flo, Vu, Sal Flo, Vu, Sal Stu, Flo, Vu, Sal 3) Who is everyone that has worked over 20 years and is aged over 50 years? Stu Mike, Kee, Stu, Flo, Vu, Sal ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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4) The company boss wants to offer an early-retirement package to everyone at the company who is aged over 50 years and has worked over 20 years. Which set operation is appropriate? Union Intersection 5) The company boss wants to give a gift to anyone who has worked over 20 years or is aged over 50 years. Which set operation is appropriate? Union Intersection 6) The company boss makes a rule: IF a person has worked over 20 years OR a person is aged over 50 years, THEN the person should get a gift. Who should get a gift? Mike, Kee, Flo, Vu, Sal Mike, Kee, Stu, Flo, Vu, Sal 7) The company boss wants to know who has worked over 20 years but is not aged over 50 years. Who are those people? Mike, Kee Mike, Kee, Stu Flo, Vu, Sal Example: Cold and ±u Venn diagrams The Venn diagram below was published by a government health agency. The diagram quickly helps determine whether a person's symptoms suggest a cold or ²u (in²uenza). Notice how much more understandable the diagram is versus similar descriptions in textual or tabular form. Figure 4.10.1: Venn diagram of cold and ²u symptoms. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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PARTICIPATION ACTIVITY 4.10.4: Venn diagram for healthy behaviors. Consider the Venn diagram above. 1) How many sets are shown? 1 2 2) In the sets labeled Cold and In²uenza, what are the members of each set? The people who have that illness The symptoms of each illness 3) Runny nose is a symptom of which illness? Cold only In²uenza only Both cold and in²uenza 4) Sore throat is a symptom of which illness? Cold only In²uenza only Both cold and in²uenza Depicting overlapping sets Source: Dept. of Health and Human Services, Montana ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Venn diagrams are commonly used to depict overlapping sets of items without actually showing the set members. PARTICIPATION ACTIVITY 4.10.5: Venn diagrams commonly show sets without showing members. PARTICIPATION ACTIVITY 4.10.6: Venn diagram for sets without showing members. Consider the Venn diagrams above. 1) Considering the Venn diagram on the left, how many applicants have the required technical skills and good social skills? About 15 About 15% Unknown 2) Considering the Venn diagram on the right, does a region exist for people who have technical skills and experience, but lack social skills? Yes No 3) How many initial sets are shown on the right? 1 2 3 Animation content: unde±ned Animation captions: 1. Venn diagrams can just depict sets without showing members. Here, a hiring manager depicts that some applicants have technical skills and others have social skills. 2. The manager wants people to understand that the ideal hires are people who are members of both sets. The Venn diagram helps people understand the manager's point. 3. Venn diagrams are commonly used for more than two sets. Here, the manager adds a third set: People having experience. The ideal hire is a member of all three sets. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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Venn diagrams for humor Venn diagrams are sometimes used to make a point in a humorous way. Below is a commonly-shared Venn diagram related to social media posting. Venn diagrams with data Sometimes a Venn diagram is annotated with data to indicate how many items fall into each region. The ±gure below is from an article summarizing people who follow common healthy-living recommendations, namely: avoiding smoking, eating fruits and vegetables, and doing physical activity. Figure 4.10.2: Venn diagram of healthy behaviors in Americans. Source: CDC The reader is encouraged to notice how elegantly the Venn diagram depicts the three sets of people, and how easily various questions about the groups can be answered, such as "What percentage of people eat fruits/vegetables but also smoke and don't do physical activity?" The answer is easily seen to be 2%: The portion of the left circle that is outside the other two circles. PARTICIPATION ACTIVITY 4.10.7: Venn diagram for healthy behaviors. Consider the Venn diagram above. ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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1) What percentage of people abstain from smoking? 47% 76% 100% 2) What percentage of people do all three healthy behaviors: abstain from smoking, eat fruits/vegetables, and do physical activity? 5% 14% 100% 3) What percentage of people eat fruits/vegetables and do physical activity, but smoke? 1% 6% No such people exist 4) What percentage of people do at least one of the three healthy behaviors? 83% 100% 118% 5) What percentage of people don't do any of the three healthy behaviors? 1% 18% ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467 ©zyBooks 11/03/23 16:51 1490122 Darian Muzny MTH_216T_57303467

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CH 4

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