Homework 1 Solutions

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1 Homework 1 Solutions Chapter 1B Multiple Negations. Explain the meaning of the given statement, then answer the question that follows. 23. Sarah did not decline the offer to go to dinner. Did Sarah go to dinner? Sarah accepted the offer to go to dinner, so Sarah did go to dinner. And Statements. The following propositions have the form p and q. State p and q, and give their truth values. Then determine whether the entire proposition is true or false, and explain why. 35. Some people are happy and some people are short. Some people are happy is a true statement, as is the statement some people are short, so the conjunction is true. Interpreting or. State whether or is used in the inclusive or exclusive sense in the following propositions. 41. My next vacation will be in Mexico or Costa Rica. This is an example of an exclusive or (assuming the planned vacation is not some kind of tour). If... then Statements. Identify the hypothesis and conclusion in the following propositions, and state their truth values. Then determine whether the entire proposition is true or false. 67. If pigs can fly, then fish can brush their teeth. The hypothesis is pigs can fly and the conclusion is fish can brush their teeth ; since the hypothesis is false, the entire proposition is vacuously true. Chapter 1C Classifying Numbers. Choose the first set in the list natural numbers, whole numbers, integers, rational numbers, and real numbers that describes the following numbers /3 rational number 16. 5/2 rational number 19. π real number /456 rational number

2 Set Notation. Use set notation (braces) to write the members of the following sets, or state that the set has no members. You may use... to indicate patterns. 32. Every third number between 4 and 20 beginning with 4 {4, 7, 10, 13, 16, 19} 33. The perfect squares (such as 1 2, 2 2, 3 2 ) between 30 and 130 {6 2, 7 2, 8 2, 9 2, 10 2, 11 2 } = {36, 49, 64, 81, 100, 121} 34. The kings of (the U.S. of) America The set is empty. 35. Odd numbers between 2 and 30 that are multiples of 3 {3, 9, 15, 21, 27} Venn Diagrams for Two Sets. Draw Venn diagrams with two circles showing the relationship between the following pairs of sets. Provide an explanation of the diagram you drew. 39. words and verbs 40. reptiles and bacteria words verbs reptiles bacteria Categorical Propositions. For the given categorical propositions, do the following. a. If necessary, rephrase the statement in standard form. b. State the subject and predicate sets. c. Draw a Venn diagram for the propositions and label all regions of the diagram. 47. All U.S. presidents have been over 30 years old. a. All U.S. presidents are people over 30 years old. b. S = U.S. presidents; P = people over 30 years old people over 30 U.S. presidents c.

3 48. Every child can sing. a. All children are people who can sing. b. S = children; P = people who can sing people who can sing children c. Venn Diagrams for Three Sets. Draw Venn diagrams with three overlapping circles (eight regions) for the following groups of three sets. Describe the members of each region or state that a region has no members. 56. oceans, bodies of salt water, bodies of fresh water fresh & salt water ocean = empty salt water oceans oceans salt water bodies fresh water bodies fresh water ocean = empty fresh & salt body of water Venn Diagram with Numbers. Use the Venn diagram to answer the following questions. people at a party men under age a. How many women at the party are under 30? 16 b. How many men at the party are not under 30? 22 c. How many women are at the party? 44 d. How many people are at the party? 81

4 Venn Diagram with Numbers. Use the Venn diagram to answer the following questions. people at a conference women college degree currently employeed a. How many people at the conference are employed men without a college degree? 6 b. How many people at the conference are unemployed women? 24 c. How many people at the conference are unemployed men without a college degree? 3 d. How many people are at the conference? Readership Survey. A (hypothetical) survey revealed the following results about the news sources that a sample of 130 people use: TV/radio only 20 TV/radio and Internet only 12 Internet only 29 TV/radio and newspapers only 18 Newspapers only 15 Internet and newspapers only 22 None 6 All three sources 8 a. Draw a three circle Venn diagram that summarizes the results of the survey. TV/radio Internet Newspapers 15 6 b. How many people use (at least) TV/radio and newspapers? 26 c. How many people use TV/radio or Internet? 109 d. How many people use TV/radio or Internet, but not newspapers? 61 e. How many people use Internet, but not TV/radio? 51 f. How many people use TV/radio, but not newspapers? 32

5 Organizing Propositions. Draw a Venn diagram that represents the information in the following statements. Use the diagram (and no other information) to answer the questions that follow. Explain your reasoning. 83. All meat has protein. All dairy products have protein. Some beans have protein. All beans, but no meat or dairy products, are plants. Questions: Could there be beans that are dairy products? Could there be meat that is a dairy product? Could there be dairy products that are plants? Could there be plants with protein? meat protein dairy plants beans No beans can be dairy products (since no dairy product are plants); Some meat could be a dairy product; No dairy products are plants (as given); and, some plants have protein (e.g. some beans). 84. No Republicans are Democrats. No Republicans are Green Party members. All Republicans are conservative. Some liberals are Democrats. No liberals are conservatives. Questions: Could there be conservative Democrats? Party members? Could there be liberal Republicans? Could there be liberal Green liberals Dem GP conservatives Rep Yes, there can be conservative Democrats; Yes, there can be liberal Green Party members; but, since all Republicans are conservative and no liberals are conservative, there cannot be liberal Republicans. Chapter 2A Working with Fractions. The following exercises require the skills covered in the Brief Review on pp Write each of the following as a common fraction. a. 3.5 b. 0.3 c d. 4.1 e f g h a. 7 / 2 b. 3 / 10 c. 1 / 20 d. 41 / 10 e. 43 / 20 f. 7 / 20 g. 49 / 50 h. 401 / 100

6 17. Convert the following fractions to decimal form; round to the nearest thousandth if necessary. 1 3 a. 4 b. 8 c d. 5 e f g. 50 h. 26 a b c d. 0.6 e. 6.5 f g h Identifying Units. Identify the units of the following quantities. State the units mathematically (for example, mi/hr) and in words (for example miles per hour). 21. The cost of a piece of carpet, found by dividing the price in dollars by its area in square yards dollars per square yard (USD/yd 2 ) 22. The flow rate of a river in which 5000 cubic feet of water flow past a particular location every second 5000 cubic feet per second (ft 3 /s) Unit Conversions. Carry out the following unit conversions. 27. Convert 24 feet to inches. 12 in 24 ft = 24 ft = 288 in 1 ft 28. Convert 24 feet to yards. 24 ft = 24 ft 1 yd 3 ft = 8 yd Conversions with Square and Cubic Units. 39. Find a conversion factor between square feet and square inches. Write it in three forms. 1 ft 2 = 12 in 12 in = 144 in 2 ; 144 in 2 /1 ft 2 = 1; 1 ft 2 /144 in 2 = Find the area in square feet of a rectangular yard that measures 20 yards by 12 yards. 20 yd 12 yd = 20 yd 3 ft 3 ft 1 yd 12 yd 1 yd = 2160 ft2 Currency Conversions. Use the currency exchange rates in Table 2.1 for the following question. 49. You return from Mexico with 3000 pesos. How much are they worth in U.S. dollars? USD 3000 pesos = USD 1 peso

7 Working with Units. Use unit conversions to answer the following questions. 55. An airliner travels 45 miles in 5 minutes. What is its speed in miles per hour? 45 mi 60 min = 540 mi 5 min 1 hr hr 56. What is the total cost of 1.2 cubic yards of soil if it sells for $24 per cubic yard? 1.2 yd 3 $24 yd 3 = $28.80 Gas Mileage. Answer the following practical gas mileage questions. 75. Gas mileage actually varies slightly with the driving speed of a car (as well as with highway vs. city driving). Suppose your car averages 38 miles per gallon on the highway if your average speed is 55 miles per hour, and it averages 32 miles per gallon on the highway if your average speed is 70 miles per hour. a. What is the driving time for a 2000-mile trip if you drive at an average speed of 55 miles per hour? What is the driving time at 70 miles per hour? 55 mi mi 2000 mi hr; 2000 mi hr 1 hr 1 hr b. Assume a gasoline price of $2.90 per gallon. What is the gasoline cost for a mile trip if you drive at an average speed of 55 miles per hour? What is the gasoline cost at 70 miles per hour? ( 2000 mi 38 mi 1 gal ) $ gal $152.63; ( 2000 mi ) 32 mi $ gal 1 gal = $ Suppose your car averages 32 miles per gallon on the highway if your average speed is 60 miles per hour, and it averages 25 miles per gallon on the highway if your average speed is 75 miles per hour? a. What is the driving time for a 1500-mile trip if you drive at an average speed of 60 miles per hour? What is the driving time at 75 miles per hour? 60 mi mi 1500 mi = 25 hr; 1500 mi 75 = 20 hr 1 hr 1 hr b. Assume a gasoline price of $2.90 per gallon. What is the gasoline cost for a mile trip if you drive at an average speed of 60 miles per hour? What is the gasoline cost at 75 miles per hour? ( 1500 mi 32 mi 1 gal ) $ gal $135.94; ( 1500 mi ) 25 mi $ gal 1 gal = $ Shower vs. Bath Assume that when you take a bath, you fill a tub to the halfway point and the tub measures 6 feet by 3 feet by 2.5 feet. When you take a shower, you use a shower head with a flow rate of 1.75 gallons per minute, and you typically spend 10 minutes in the shower. There are 7.5 gallons in one cubic foot. a. Do you use more water taking a shower or taking a bath?

8 bath = 1 / 2 (6 ft 3 ft 2.5 ft) = 22.5 ft 3 shower = 1.75 gal min is greater than 1 ft3 10 min 2.33 ft3 7.5 gal b. How long would you need shower in order to use as much water as you use taking a bath? ( 1.75 gal 22.5 ft 3 min 1 ) ft3 96 min 7.5 gal c. Assuming your shower is in a bath tub, propose a nonmathematical way to compare, in one experiment, the amounts of water you use taking a shower and a bath. You could simply plug the drain in the tub and take a shower as normal.