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. 15. Negate both p and r and find the disjunction. A disjunction is true if at least one of the statements is ~p is is not the angle bisector of, which is false. ~r is is not congruent to, which is false. Thus, ~p or ~r is false because ~p is false and ~r is false. 11. p and r p and r is a conjunction. A conjunction is true only when both statements that form it are p is the angle bisector of, which is r is, which is Thus, p and r is true because p is true and r is is the angle bisector of and. p and r is true because p is true and r is is is not the angle bisector of, or. ~p or ~r is false because ~p is false and ~r is false. 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. 13. Negate p, then find the disjunction. A disjunction is true if at least one of the statements is r is true since. The negation of p is is not the angle bisector of, which is false. Thus, r or ~p is true because r is is not the angle bisector of. r or ~p is true because r is 17. is a conjunction. A conjunction is true only when both statements that form it are p is Austin is the capital of Texas, which is r is Texas shares a border with Louisiana, which is Then is true, because p is true and r is Austin is the capital of Texas, and Texas shares a border with Louisiana. is true because p is true and r is esolutions Manual - Powered by Cognero Page 1
19. Negate r and find the disjunction. A disjunction is true if at least one of the statements is ~r is Texas does not share a border with Louisiana, which is false. s is Texas is west of New Mexico, which is false. Then is false, because ~r is false and s is false. 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. 23. Copy and complete each truth table. Add values to the column for q so that each pair of p and q are distinct. ~p q is a conjunction. A conjunction is true only when both statements that form it are ~p q is only true when ~p is true and q is 21. Negate both p and r and find the conjunction. A conjunction is true only when both statements that form it are ~p is Austin is not the capital of Texas, which is false. ~r is Texas does not share a border with Louisiana, which is false. Then is false because ~p is false and ~r is false. 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. esolutions Manual - Powered by Cognero Page 2
Construct a truth table for each compound statement. 25. Write the truth values T and F for p and r so that each pair is distinct. p r is a conjunction. A conjunction is true only when both statements that form it are Then p r will only be true with both p and r are 29. Add truth values T and F for p and r so that each pair is distinct. Negate p. This means, find the opposite truth value. ~p r is a conjunction. A conjunction is true only when both statements that form it are ~p r will be true only when both ~p and r are 27. Add truth values T and F for p and r so that each pair is distinct. p r is a disjunction. A disjunction is true if at least one of the statements is p r will be true for all values except when both r and p are false. esolutions Manual - Powered by Cognero Page 3
31. WATER SPORTS Refer to the Venn diagram that represents the number of students who swim and dive at a high school. 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. a. How many students dive? b. How many students participate in swimming or diving or both? c. How many students swim and dive? a. The number of students who dive is listed in the dive circle. Three students participate in both swimming and diving. Four students participate only in diving. So, 7 students participate in diving. b. To find the students that do either or both, find the union of the sets. The number of students who participate in swimming or diving or both is 19 + 3 + 4 or 26. c. To find the number of students that participate in both activities, find the intersection. The number of students who participate in both activities is 3. b. Since the light is on when the upstairs switch is up and the downstairs switch is down, when both switches are up, the value is false in the light on column. c. Since the light is on when the upstairs switch is up and the downstairs switch is down 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. a. a. 7 b. 26 c. 3 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. 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. b. If both the upstairs and downstairs switches are in the up position, will the light be on? Explain your reasoning. esolutions Manual - Powered by Cognero Page 4
35. Construct a truth table for each compound statement. Determine the truth value of each compound statement if the given statements are is the conjunction of p with the disjunction of ~q and r. First negate q. Then find the disjunction with r. The disjunction will be true when either ~q or r are Then find the conjunction with p. It will be true with both p and is 37. Negate both q and r, finding the opposite truth values. Then find the conjunction ~q and ~r. The conjunction will be true when both ~q and ~r are Then find the disjunction. It will be true if either p or is If p, q, and r are true, then the given statement is If p and r are true, then the statement is If p and r are true, then is If p, q, and r are true, then is 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. Integers are never irrational numbers. Both belong to the real number, but they never intersect. Integers are rational numbers, not irrational. Never; integers are rational numbers,not irrational. esolutions Manual - Powered by Cognero Page 5
49. ORGANIZE IDEAS Write a compound statement that results in a true conjunction. For a conjunction to be true, both statements must be A triangle has three sides, and a square has four sides. Both are true, so the compound statement is A triangle has three sides, and a square has four sides. Both are true, so the compound statement is 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. ~p and q: The formula for the volume of a sphere is not, and the formula for the surface area of a sphere is. Although q is true, ~p is false, so ~p and q is false. p or ~q: The formula for the volume of a sphere is, or the formula for the surface area of a sphere is not. Since p is true, p or ~q is Choice E is correct. E 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 Write each compound statement in words and then find its truth value. ~p or ~q: The formula for the volume of a sphere is not, or the formula for the surface area of a sphere is not false, ~p or ~q is false.. Since both ~p and ~q are ~p and ~q: The formula for the volume of a sphere is not, and the formula for the surface area of a sphere is not. Since both ~p and ~q are false, ~p and ~q is false. p and ~q: The formula for the volume of a sphere is, and the formula for the surface area of a sphere is not. Although p is true, ~q is false, so p and ~q is false. esolutions Manual - Powered by Cognero Page 6
51. Sonia made the Venn diagram below to help her remember volume formulas. Which of the following figures would be located in the shaded part of the Venn diagram? F Cone G Cylinder H Prism J Pyramid For the figure to be within the shaded region of the Venn diagram, the figure would have as a factor, but would not have as a factor. Analyze the volume formulas for each answer choice. Cone: ; the cone has a factor of and a factor of, so cone would not be within the shaded region of the Venn diagram. Cylinder: ; the cylinder does not have a factor of and it does have a factor of, so cylinder would not be within the shaded region of the Venn diagram. Prism: ; the prism does not have a factor of or a factor of, so prism would not be within the shaded region of the Venn diagram. Pyramid: ; the pyramid has a factor of and does not have a factor of, so pyramid would be in the shaded region of the Venn diagram. The correct answer is Choice J. J esolutions Manual - Powered by Cognero Page 7