2-2 Logic ANSWER: A week has seven days, and there are 20 hours in a day. is false, because q is false. 3. ANSWER:

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1 Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. Explain your reasoning. p : A week has seven days. q: There are 20 hours in a day. r: There are 60 minutes in an hour. 1. p and r A week has seven days, and there are 60 minutes in an hour. p and r is true, because p is true and r is 7. Copy and complete the truth table at the right A week has seven days, and there are 20 hours in a day. is false, because q is false. Construct a truth table for each compound statement. 8. There are 20 hours in a day, or there are 60 minutes in an hour. is true, because r is 4. A week does not have seven days, or there are 20 hours in a day. ~p or q is false, because ~p is false and q is false A week has seven days, or there are 60 minutes in an hour. is true, because p is true and r is 6. A week does not have seven days, and there are not 60 minutes in an hour. is false, because ~p is false and ~r is false. esolutions Manual - Powered by Cognero Page 1

2 10. CLASSES Refer to the Venn diagram that represents the foreign language classes students selected in high school. a. How many students chose only Spanish? b. How many students chose Spanish and French? c. Describe the class(es) the three people in the nonintersecting portion of the French region chose r and q is not the angle bisector of. r or ~p is true because r is and Points C, D, and B are collinear. r and q is false because q is false. 15. a. 89 b. 15 c. The 3 students chose to take only French classes. Use the following statements and figure to write a compound statement for each conjunction or disjunction. Then find its truth value. Explain your reasoning. p : is the angle bisector of. q: Points C, D, and B are collinear. 16. is not the angle bisector of, or. ~p or ~r is false because ~p is false and ~r is false. is not the angle bisector of, and. ~p and ~r is false because ~p is false and ~r is false 11. p and r is the angle bisector of and. p and r is true because p is true and r is 12. q or p Points C, D, and B are collinear, or is the angle bisector of. q or p is true because p is esolutions Manual - Powered by Cognero Page 2

3 JUSTIFY ARGUMENTS Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. Explain your reasoning. p : Austin is the capital of Texas. q: Texas borders the Pacific Ocean. r: Texas shares a border with Louisiana. s: Texas is west of New Mexico. 21. Austin is not the capital of Texas, and Texas does not share a border with Louisiana. is false because ~p is false and ~r is false. 22. Texas is not west of New Mexico, or Austin is not the capital of Texas. is true because ~s is Copy and complete each truth table. 17. Austin is the capital of Texas, and Texas shares a border with Louisiana. is true because p is true and r is Austin is the capital of Texas, and Texas borders the Pacific Ocean. is false because q is false. 19. Texas does not share a border with Louisiana, or Texas is west of New Mexico. is false because ~r is false and s is false Texas shares a border with Louisiana, or Texas borders the Pacific Ocean. is true because r is esolutions Manual - Powered by Cognero Page 3

4 Construct a truth table for each compound statement WATER SPORTS Refer to the Venn diagram that represents the number of students who swim and dive at a high school. 28. a. How many students dive? b. How many students participate in swimming or diving or both? c. How many students swim and dive? a. 7 b. 26 c. 3 esolutions Manual - Powered by Cognero Page 4

5 32. ORGANIZE IDEAS Venus has switches at the top and bottom of her stairs to control the light for the stairwell. She notices that when the upstairs switch is up and the downstairs switch is down, the light is turned on. a. Copy and complete the truth table. b. If both the upstairs and downstairs switches are in the up position, will the light be on? Explain your reasoning. c. If the upstairs switch is in the down position and the downstairs switch is in the up position, will the light be on? d. In general, how should the two switches be positioned so that the light is on? a. b. No; when both switches are up, the value is false in the light on column. c. Yes; when the upstairs switch is down and the downstairs switch is up, the value is true in the light on column. d. The light is on when switches are in opposite positions. 33. MULTI-STEP A group of 370 teens were surveyed about what type of electronics they wanted to purchase. They chose from a smartphone, a digital media player, and a DVR. The results are shown in the Venn diagram. a. Make 5 conclusions based on the provided data. b. If an electronics store owner already has 65 digital media players, 25 DVRs, and 100 smartphones in stock, how might he adjust his inventory based on this data? c. What assumptions did you make? What real-world constraints did you consider? d. What additional information would allow you to draw more accurate conclusions? a. Sample answer: 50 teens wanted to purchase a digital media player and a DVR. 40 teens wanted to purchase all three electronic devices. 110 teens wanted to purchase a smartphone only. 20 teens wanted to purchase a smartphone and a digital media player. 10 teens did not want to purchase any electronics. b. Sample answer: The electronics store owner may consider increasing the quantity of these items that he has in stock. However, he probably will not purchase enough electronics to meet the current demand. c. Sample answer: I assumed that the teens desire to purchase the electronics guaranteed an increase in the number of electronics that the store owner expects to sell. However, I also assumed that the store owner does not expect to sell all of the electronics that the teens want to purchase. I considered that a desire to purchase does not guarantee that the teens have the means to purchase. The teens may choose to buy a portion of the electronics at another store. I also considered that the product brand carried by the store owner might not be the type of product that each teen wants to purchase. d. It would help to know how each teen prioritizes their desired purchases if they want to purchase multiple electronic devices. What product brand each esolutions Manual - Powered by Cognero Page 5

6 34. student wants would also help influence the store owner s purchases. Construct a truth table for each compound statement. Determine the truth value of each compound statement if the given statements are If p and q are true, then the statement is true if r is true and false if r is false. 37. If p, q, and r are true, then is 38. If p, q, and r are true, is false. 35. If p and r are true, then is 39. If p and q are true, then is 36. If q and r are both true, then is esolutions Manual - Powered by Cognero Page 6

7 40. ANALYZE RELATIONSHIPS A San Antoniobased hotel surveyed 95 of their guests about their vacation plans during their stay. Of the 95 guests who had visited San Antonio, 75 had visited The Alamo, Brackenridge Park, or both. Of those 75 guests, 55 had visited The Alamo, and 45 had visited Brackenridge Park. a. Make a Venn diagram to show the results of the survey. b. If p represents a guest who has visited The Alamo and q represents a guest who has visited Brackenridge Park, write a compound statement to represent each area of the Venn diagram. Include the compound statements on your Venn diagram. c. What is the probability that a randomly chosen participant in the survey will have visited both The Alamo and Brackenridge Park? Explain your reasoning. a. b. 41. ANALYZE RELATIONSHIPS Irrational numbers and integers both belong to the set of real numbers (R). Based upon the Venn diagram, is it sometimes, always, or never true that integers (Z) are irrational numbers (I)? Explain your reasoning. Never; integers are rational numbers,not irrational. JUSTIFY ARGUMENTS To negate a statement containing the words all or for every, you can use the phrase at least one or there exists. To negate a statement containing the phase there exists, you can use the phrase for all or for every. p : All polygons are convex. ~p: At least one polygon is not convex. q: There exists a problem that has no solution. ~q: For every problem, there is a solution. Sometimes these phrases may be implied. For example, The square of a real number is nonnegative implies the following conditional and its negation. p : For every real number x, x 2 0. ~p: There exists a real number x such that x 2 < 0. Use the information given to write the negation of each statement. 42. Every student at Hammond High School has a locker. There exists at least one student at Hammond High School who does not have a locker. c. ; Sample answer: Since 25 of those surveyed visited both The Alamo and Brackenridge Park and there were 95 survey participants, the probability that a randomly chosen participant would have visited both The Alamo and Brackenridge Park is. 43. All squares are rectangles. There exists at least one square that is not a rectangle. 44. There exists a real number x such that For every real number x, x 2. esolutions Manual - Powered by Cognero Page 7

8 45. There exists an even number x such that. For every even number x,. 46. Every real number has a real square root. There exists a real number that does not have a real square root. 47. There exists a segment that has no midpoint. Every segment has a midpoint. 48. WRITING IN MATH Describe a situation that might be depicted using the Venn diagram. 50. ACT/SAT Consider the statements below. p : The formula for the volume of a sphere is. q: The formula for the surface area of a sphere is. Which of the following compound statements is true? A ~p or ~q B ~p and ~q C p and ~q D ~p and q E p or ~q E 51. Sonia made the Venn diagram below to help her remember volume formulas. Sample answer: 100 people were surveyed to see if they liked vanilla, strawberry, or chocolate ice cream. There were 8 people who only liked strawberry. There were 25 people who liked both strawberry and vanilla. There were 48 people who only liked vanilla, and there were 19 people who liked both chocolate and vanilla. 49. ORGANIZE IDEAS Write a compound statement that results in a true conjunction. A triangle has three sides, and a square has four sides. Both are true, so the compound statement is Which of the following figures would be located in the shaded part of the Venn diagram? F Cone G Cylinder H Prism J Pyramid J esolutions Manual - Powered by Cognero Page 8