HW #3-SOLUTIONS Topics: Sets, categorical propositions, Venn diagrams, analyzing arguments, and critical thinking Please show your work and clearly indicate your answer. Although you are welcome to compare your methods and answers with other students, be careful not to copy off of other students. When indicated you may use a calculator, otherwise only do so to check your work. I. Sets & Venn diagrams [50] 1. Sets. Use set notation to write the set of months in the year. [2 points] {September, October, November, December, January, February, March, April, May, June, July, August} 2. Venn diagram. Draw a Venn diagram representing the relationship between the following sets. Provide an example that would go in each region. [3 points each] a. Famous Singers and Famous Mathematicians Singers (Madonna) Mathematicians (Fibonacci) b. Animals that fly and Pigeons Animals that Fly (Owls) Pigeons (Carrier Pigeons) c. Devices that use electricity and devices that can go underwater Devices that use electricity (hair dryers) Devices that can go underwater (fishing nets) submarines IGE104: Logic and Mathematics for Daily Living (3-2010) Page 1 of 8
3. Categorical propositions. Rewrite the following categorical propositions in standard form, and draw a Venn diagram representation. [4 points each] a. Every bird can sing All are creatures that sing: Bird s Creatures that sing b. No worms eat No worms are creatures that eat : Creatures that eat Worms c. Some cannot fly Some are not creatures that fly Creatures that fly Birds d. Some can fly Some are creatures that fly: Creatures that fly Birds 4. Venn diagram. Draw a Venn diagram that represents the relationship between the following three sets: words that begin with S, words that end in E, and people s names. Provide an example that would belong to each region of your Venn diagram. [5 points] some Words that begin with S super Words that end in E possible Sammie Sarah People s Names Kim Emily IGE104: Logic and Mathematics for Daily Living (3-2010) Page 2 of 8
5. Venn diagram. Draw a Venn diagram for the data; include the number of observations in each region. [6 points] (number of people) No oral cancer Oral Cancer One tomato per day 191 9 Fewer than three tomatoes per week 164 16 One tomato per day (191) Oral Cancer (9) (16) <3 tomatoes per week (164) 6. Categorical propositions. Consider the following statements: All meat has protein. All dairy products have protein. Some beans have protein. All beans, but no meat or dairy products are plants. a. Draw a Venn diagram that represents the information in the statements. [4 points] Protein Meat Beans Plants Dairy b. Could there be beans that are dairy products? No. From the statement: all beans, but no meat or dairy products are plants so we know that Beans are a subset of Pants, while plants and dairy products are disjoint. Thus it is impossible for any members of Beans to belong to the set Dairy. c. Could there be meat that is a dairy product? Yes. From the statement: all dairy are protein and all meat are protein so we know that both dairy and meat are subsets of the set Protein. However the statement does not describe the relationship between Dairy and Meat, so it is possible that some members of Meat belong to the set Dairy. d. Could there be dairy products that are plants? No. From the statement: all beans, but no meat or dairy products are plants so we know that dairy products are not plants. e. Could there be plants with protein? Yes. From the statement: all beans, but no meat or dairy products are plants we know that both beans are a subset of Plants. We also know from some beans are protein that some members of Beans (and therefore members of Plants are also members of Protein. IGE104: Logic and Mathematics for Daily Living (3-2010) Page 3 of 8
II. Analyzing Arguments [42] 7. Identify if the following arguments are inductive or deductive. [2 points each] a. I have never found mail in my mailbox on a Sunday. The Postal Service must not have Sunday deliveries Inductive since not finding mail in the mailbox every Sunday can be viewed as a specific observation, while the conclusion the Postal Service must not have Sunday deliveries is a specific conclusion. b. Because of a budget cutback, postal workers will no longer work on Saturdays. Therefore, I will not expect Saturday deliveries in the future. Deductive since the premise is a specific statement regarding the work of postmen and the conclusion I will not expect Saturday deliveries in the future is general. 8. Inductive. Determine the truth of the premises, and assess the strength of the argument. Is the conclusion true? [4 points] Premise: Cats and dogs have hair and they are mammals. (T) Premise: Mice and rats have hair and they are mammals. (T) Premise: Tigers and lions have hair and they are mammals. (T) Conclusion: Animals with hair are mammals. The argument seems strong since it has three true supporting premises each with two examples of mammals that have hair. However considering the sheer number of animals that are classified as mammals perhaps more examples should be provided. 9. Inductive. Determine the truth of the premises, and assess the strength of the argument. Is the conclusion true? [4 points] Premise: Bach, Buxtehude, Beethoven, Brahms, Berlioz, and Britten are great composers. (T) Conclusion: Composers with names that begin with B are great. The argument provides a list of 6 great composers whose names begin with a B, however, relative to the number of composers that existed throughout history, 6 is not a very large number. Using common sense we can also see that talent has nothing to do with the first letter of someone s last name, so it seems unlikely that all composers whose names begin with B are great. Thus for this argument to be strong enough to overcome our common sense intuition regarding the conclusion, we would need many more examples of great composers whose names begin with B. For questions 10-14 consider the arguments below. For each one: a. If necessary, rephrase the first premise so that it has the form all S are P (for categorical) or the form if p then q (for conditional). b. Draw a Venn diagram to determine whether the argument is valid. [3 points] c. Discuss the truth of the premises (you may need to do a quick online search) and state whether the argument is sound. [3 points] IGE104: Logic and Mathematics for Daily Living (3-2010) Page 4 of 8
10. Deductive/Categorical. Premise: All European countries use the euro as currency (1) Premise: Great Britain is a European country. (2) Conclusion: Great Britain uses the euro as currency. a. Premise (1): All European countries are countries that use the Euro as currency. b. Premise (1): determines the relative positions of the two sets in the Venn diagram Premise (2): marks an representing Great Britain within European Countries Countries that use the Euro as currency European Countries From the Venn diagram above, the conclusion Great Britain uses the euro as currency is valid. c. Premise (1) says All European countries use the euro as currency is not true, since there are many countries that are in the continent of Europe that do not use the Euro. The second premise is true. However, since Premise (1) is false, the argument is not sound. 11. Deductive/Categorical. Premise: All fruit is fat-free. (1) Premise: Avocados are fruit. (2) Conclusion: Avocados are fat-free. a. Premise (1): All fruit are fat-free foods. b. Premise (1): determines the relative positions of the two sets in the Venn diagram Premise (2): marks an representing Avocados within Fruit Fat-free foods Fruit From the Venn diagram above, the conclusion Avocados are fat-free is valid. c. Premise (1) says All fruit is fat free is not true, since oil is a type of fat (that is liquid at room temperature) and fruit such as coconut, olive, palm, and even avocado store oil in their seeds. The second premise is true since Avocado are indeed fruit of the Avocado tree (Persea americana). However, since Premise (1) is false, the argument is not sound. IGE104: Logic and Mathematics for Daily Living (3-2010) Page 5 of 8
12. Deductive/Conditional. Premise: If it is a bird, then its young are hatched from eggs. (1) Premise: Eagles are. (2) Conclusion: Eagles chicks are hatched from eggs. a. Premise (1): (ok) b. Premise (1): determines the relative positions of the two sets in the Venn diagram Premise (2): marks an representing Eagles within It is a bird Its young are hatched from eggs It is a bird From the Venn diagram above, the conclusion Eagles chicks are hatched from eggs is valid. c. Premise (1) says If it is a bird, then its young are hatched from eggs is true, since for an animal to be classified as a bird it must lay eggs. The second premise is also true since Eagles are indeed a type of bird (belonging to the Family Accipitridae). Since both Premise (1) and Premise (2) are true, the argument is sound. 13. Deductive/Conditional. Premise: If we put a man on the Moon, we can build a computer operating system that works. Premise: We can build a computer operating system that works. Conclusion: We can put a man on the Moon. a. Premise (1): (ok) b. Premise (1): determines the relative positions of the two sets in the Venn diagram Premise (2): marks an representing We can build a computer operating system that works on the border of the subset we put a man on the moon We can build a computer operating system that works We put a man on the moon From the Venn diagram above, the conclusion We can put a man on the moon is not valid. c. Since the argument structure is invalid, it doesn t matter if Premise (1) and Premise (2) are true or false, since in either case the argument is not sound. IGE104: Logic and Mathematics for Daily Living (3-2010) Page 6 of 8
14. Deductive/Conditional. Premise: If taxes are increased, then taxpayers will have less disposable income. Premise: If taxpayers have less disposable income, spending will decrease and the economy will slow down. Conclusion: If taxes are increased, the economy will slow down. b. Premise (1): determines the relative positions of sets A and B in the Venn diagram Premise (2): determines the relative positions of sets B and C in the Venn diagram C: Spending will decrease & the economy will slow down B: taxpayers will have less disposable income A: Taxes are increased From the Venn diagram above, the conclusion If taxes are increased the economy will slow down is valid because set A is a subset of set C. c. Both Premise (1) and Premise (2) are true, and since the argument is valid, the argument is sound. III. Critical Thinking [8] 15. Use critical thinking skills to analyze the argument below taken from Unpaid Interns, Complicit Colleges an Op-Ed by Ross Perlin published April 2, 2011 in The New York Times. Consider the implied conclusion of the article and the validity and truth of its premises. [8 points] Colleges and universities have become cheerleaders and enablers of the unpaid internship boom, failing to inform young people of their rights or protect them from the miserly calculus of employers While higher education has tried to stand for fairness in the past few decades through affirmative action and financial aid, the internship boom gives the well-to-do a foot in the door while consigning the less well-off to dead-end temporary jobs. Colleges have turned internships into a prerequisite for the professional world but have neither ensured equal access to these opportunities, nor insisted on fair wages for honest work. The full text of the article can be read at http://www.nytimes.com/2011/04/03/opinion/03perlin.html IGE104: Logic and Mathematics for Daily Living (3-2010) Page 7 of 8
BONUS [10] What is the conclusion that you reach after the premises below? We can break up this argument into three premises: Premise (1): All humming () are richly colored Richly colored Premise (2): No large live on honey Large Birds that live on honey (or alternatively) Birds that do not live on honey Large Premise (3): Birds that do not live on honey are dull in color. Birds that are dull in color Birds that do not live on honey Putting it all together, and noting that dull color is the opposite of richly colored we have the following diagram Birds that are dull in color Birds that do not live on honey Large Conclusion: Humming are small that live on honey. IGE104: Logic and Mathematics for Daily Living (3-2010) Page 8 of 8