2-2 Statements, Conditionals, and Biconditionals

Use the following statements to write acompound statement for each conjunction ordisjunction. Then find its truth value. Explainyour 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

SOLUTION:

p and r is a conjunction. A conjunction is true onlywhen both statements that form it are true. p is truesince a week has seven day. r is true since there are60 minutes in an hour. Then p and r is true,because both p and r are true.

ANSWER:

A week has seven days, and there are 60 minutes inan hour. p and r is true, because p is true and r istrue.

2.

SOLUTION:

is a conjunction. A conjunction is true onlywhen both statements that form it are true. p is truesince a week has seven days. q is false since thereare 24 hours in a day, not 20 hours in a day. Thus,

is false, because q is false.

ANSWER:

A week has seven days, and there are 20 hours in aday. is false, because q is false.

3.

SOLUTION:

is a disjunction. A disjunction is true if at leastone of the statements is true. q is false, since thereare 24 hours in a day, not 20 hours. r is true sincethere are 60 minutes in an hour. Thus, is true,because r is true.

ANSWER:

There are 20 hours in a day, or there are 60 minutesin an hour.

is true, because r is true.

4.

SOLUTION:

~p is a negation of statement p, or the opposite ofstatement p. The or in ~p or q indicates adisjunction. A disjunction is true if at least one of thestatements is true. ~p would be : A week does not have seven days,which is false. q is false since there are 24 hours ina day, not 20 hours in a day. Then ~p or q is false,because both ~p and q are false.

ANSWER:

A week does not have seven days, or there are 20hours in a day. ~p or q is false, because ~p is falseand q is false.

5.

SOLUTION:

is a disjunction. A disjunction is true if at leastone of the statements is true. p is true since a week has seven days. r is true,since there are 60 minutes in an hour.Thus, istrue, because p is true and r is true.

ANSWER:

A week has seven days, or there are 60 minutes inan hour. is true, because p is true and r istrue.

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2-2 Statements, Conditionals, and Biconditionals

6.

SOLUTION:

~p and ~r is the conjunction of the negations of pand r. A conjunction is true if both statements aretrue. ~p is : A week does not have seven days, which isfalse. ~r is : There are not 60 minutes in an hour,which is false. Then is false, becauseboth ~p and ~r are false.

ANSWER:

A week does not have seven days, and there are not60 minutes in an hour. is false, because~p is false and ~r is false.

Write each statement in if-then form.7. Sixteen-year-olds are eligible to drive.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis.The word then is not part of the conclusion. If you are sixteen years old, then you are eligible todrive.

ANSWER:

If you are sixteen years old, then you are eligible todrive.

8. Cheese contains calcium.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If it is cheese, then it contains calcium.

ANSWER:

If it is cheese, then it contains calcium.

9. The measure of an acute angle is between 0 and 90.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If the angle is acute, then its measure is between 0and 90.

ANSWER:

If the angle is acute, then its measure is between 0and 90.

10. Equilateral triangles are equiangular.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If a triangle is equilateral, then it is equiangular.

ANSWER:

If a triangle is equilateral, then it is equiangular.

Determine the truth value of each conditionalstatement. If true, explain your reasoning. Iffalse, give a counterexample.

11. If then x = 4.

SOLUTION:

If (–4)2 = 16. The hypothesis of theconditional is true, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false.

ANSWER:

False; if x = –4, (–4)2 =16. The hypothesis of theconditional is true, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false



12. If you live in Atlanta, then you live in Georgia.

SOLUTION:

The conditional is false. You could live in Atlanta,Georgia or Atlanta, Kansas.

ANSWER:

False; Atlanta, Kansas; The hypothesis of theconditional is true, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false.

13. If tomorrow is Friday, then today is Thursday.

SOLUTION:

The conditional statement "If tomorrow is Friday,then today is Thursday." is true. When thishypothesis is true, the conclusion is also true, sinceFriday is the day that follows Thursday. So, theconditional statement is true.

ANSWER:

True; when this hypothesis is true, the conclusion isalso true, since Friday is the day that followsThursday. So, the conditional statement is true.

14. If an animal is spotted, then it is a Dalmatian.

SOLUTION:

FalseThe animal could be a leopard. The hypothesis ofthe conditional is true, but the conclusion is false.This counterexample shows that the conditionalstatement is false.

ANSWER:

False; the animal could be a leopard. The hypothesisof the conditional is true, but the conclusion is false.This counterexample shows that the conditionalstatement is false.

15. If the measure of a right angle is 95, then bees arelizards.

SOLUTION:

The conditional statement "If the measure of a rightangle is 95, then bees are lizards." is true. Thehypothesis is false, since the measure of a rightangle is 90. A conditional with a false hypothesis isalways true, so this conditional statement is true.

ANSWER:

True; the hypothesis is false, since the measure of aright angle is 90. A conditional with a falsehypothesis is always true, so this conditionalstatement is true.

16. If pigs can fly, then 2 + 5 = 7.

SOLUTION:

The conditional statement "If pigs can fly, then 2 + 5= 7" is true. The hypothesis is false, since pigscannot fly. A conditional with a false hypothesis isalways true, so this conditional statement is true.

ANSWER:

True; the hypothesis is false, since pigs cannot fly. Aconditional with a false hypothesis is always true, sothis conditional statement is true.

17. If Jacqueline turned 14 years old last year, then shewill turn 15 this year.

SOLUTION: Hypothesis: Jacqueline turned 14 years old last yearConclusion: She will turn 15 this year 14 + 1 = 15, so if the hypothesis is true, then theconclusion is true, so the conditional is true.

ANSWER: True; the number 15 is one more than 14.



18. If a number is between 10 and 12, then it is 11.

SOLUTION: Consider all numbers between 10 and 12, 11 isbetween them, but so are 10.5 and 11.5 and manyothers.

ANSWER: False; 11.5 is between 10 and 12.

JUSTIFY ARGUMENTS Write the converse,inverse, and contrapositive of each trueconditional statement.Determine whether each related conditional istrue or false. If a statement is false, find acounterexample.

19. If a number is divisible by 4, then it is divisible by 2.

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If a number is divisible by 2, then it isdivisible by 4; false. Sample answer: 6 is divisible by2 but is not divisible by 4. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If a number is not divisible by 4, then it isnot divisible by 2; false. Sample answer: 6 is notdivisible by 4 but is divisible by 2. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If a number is not divisible by 2, thenit is not divisible by 4; true.

ANSWER:

Converse: If a number is divisible by 2, then it isdivisible by 4; false. Sample answer: 6 is divisible by2 but is not divisible by 4. Inverse: If a number is notdivisible by 4, then it is not divisible by 2; false.Sample answer: 6 is not divisible by 4 but is divisibleby 2. Contrapositive: If a number is not divisible by2, then it is not divisible by 4; true.

20. All whole numbers are integers.

SOLUTION:

If a number is a whole number, then it is an integer. The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If a number is an integer, then it is awhole number. False; Sample answer: –3. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If a number is not a whole number, then itis not an integer. False: Sample answer: –3. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If a number is not an integer, then itis not a whole number; true.

ANSWER:

If a number is a whole number, then it is an integer.Converse: If a number is an integer, then it is awhole number. False; Sample answer: –3. Inverse:If a number is not a whole number, then it is not aninteger. False: Sample answer: –3. Contrapositive: Ifa number is not an integer, then it is not a wholenumber; true.

Rewrite each statement as a biconditionalstatement. Then determine whether thebiconditional is true or false.

21. There is no school on Saturday.

SOLUTION: Write a biconditional with "if and only if". There is no school if and only if it is Saturday. This statement is false, because there is no schoolon Sundays and holidays.

ANSWER: There is no school if and only if it is Saturday; false.



22. An integer is a rational number.

SOLUTION: Write the statement as a biconditional using if andonly if. A number is an integer if and only if it is a rationalnumber; false because 0.5 is not an integer, but it isa rational number.

ANSWER: A number is an integer if and only if it is a rationalnumber; false.

23. An angle with a measure between and isacute.

SOLUTION: Make a biconditional with and "if and only if"statement An angle is acute if and only if it has a measurebetween 0 and 90 degrees; true, because thedefinition of acute is an angle between 0 and 90degrees.

ANSWER: An angle is acute if and only if it has a measurebetween and ; true.

24. An obtuse triangle has one obtuse angle.

SOLUTION: Write a biconditional using if and only if. A triangle is obtuse if and only if it has one obtuseangle; true as this is the definition of an obtusetriangle.

ANSWER: A triangle is obtuse if and only if it has one obtuseangle; true.

25. A dodecagon is a polygon with 12 sides.

SOLUTION: Write a biconditional using if and only if. A polygon is a dodecagon if and only if it has 12sides; true this is the definition of a dodecagon.

ANSWER: A polygon is a dodecagon if and only if it has 12sides; true.

26. Two angles whose measure add to arecomplementary.

SOLUTION: The parts of the statement are "two angles whosemeasures add to " and "two angles arecomplementary" A biconditional is of the form if and only if. Two angles have measures that add to if andonly if the angles are complementary. This is true, because it is the definition ofcomplementary angles.

ANSWER:

Two angles have measures that add to if andonly if the angles are complementary; true.

27. A number x such that is positive.

SOLUTION: A number x is greater than -0.5 if and only if x ispositive; false, because x could be 0.

ANSWER: A number x is greater than -0.5 if and only if x ispositive; false.



Use the following statements and figure towrite a compound statement for eachconjunction or disjunction. Then find itstruth value. Explain your reasoning.

p: is the angle bisector of .q: Points C, D, and B are collinear.

28. p and r

SOLUTION:

p and r is a conjunction. A conjunction is true onlywhen both statements that form it are true. p is is the angle bisector of , which is true. r is

, which is true. Thus, p and r is truebecause p is true and r is true.

ANSWER:

is the angle bisector of and .p and r is true because p is true and r is true.

29. q or p

SOLUTION:

q or p is a disjunction. A disjunction is true if at leastone of the statements is true. q is false since points

C, D, and B are not collinear. p is true since isthe angle bisector of . Thus, q or p is truebecause p is true.

ANSWER:

Points C, D, and B are collinear, or is the anglebisector of . q or p is true because p is true.

30.

SOLUTION:

Negate p, then find the disjunction . Adisjunction is true if at least one of the statements istrue. r is true since . The negation of p is

is not the angle bisector of , which isfalse. Thus, r or ~p is true because r is true.

ANSWER:

is not the angle bisector of . r or ~p is true because r is true.

31. r and q

SOLUTION:

r and q is a conjunction. A conjunction is true onlywhen both statements that form it are true. r is true

since . q is false, since points C, D, andB are not collinear. Thus r and q is false becauseq is false.

ANSWER:

and Points C, D, and B are collinear. rand q is false because q is false.

32.

SOLUTION:

Negate both p and r and find the disjunction. Adisjunction is true if at least one of the statements istrue.

~p is is not the angle bisector of , which

is false. ~r is is not congruent to , which isfalse. Thus, ~p or ~r is false because ~p is falseand ~r is false.

ANSWER:

is not the angle bisector of , or . ~p or ~r is false because ~p is false and

~r is false.



33.

SOLUTION:

Negate both p and r and find the conjunction. Aconjunction is true only when both statements thatform it are true.

~p is is not the angle bisector of ,

which is false. ~r is is not congruent to ,which is false. Thus, ~p and ~r is false because ~pis false and ~r is false.

ANSWER:

is not the angle bisector of , and . ~p and ~r is false because ~p is false

and ~r is false

REASONING Use the following statements towrite a compound statement for eachconjunction or disjunction. Then find its truthvalue. Explain your reasoning.p: Springfield is the capital of Texas.q: Illinois borders the Atlantic Ocean.r: Illinois shares a border with Kentucky.s: Illinois is to the west of Missouri.

34.

SOLUTION:

is a conjunction. A conjunction is true onlywhen both statements that form it are true. p is Springfield is the capital of Illinois, which is true.r is Illinois shares a border with Kentucky, which istrue. Then is true, because p is true and r istrue.

ANSWER:

Springfield is the capital of Illinois, and Illinoisshares a border with Kentucky. is truebecause p is true and r is true.

35.

SOLUTION:

is a conjunction. A conjunction is true onlywhen both statements that form it are true. p is Springfield is the capital of Illinois, which is true.q is Illinois borders the Atlantic Ocean, which isfalse. Then. is false because q is false.

ANSWER:

Springfield is the capital of Illinois, but Illinois doesnot border the Atlantic Ocean. is falsebecause q is false.



36.

SOLUTION:

Negate r and find the disjunction . Adisjunction is true if at least one of the statements istrue. ~r is Illinois does not share a border withKentucky, which is false. s is Illinois is west ofMissouri, which is false. Then is false,because ~r is false and s is false.

ANSWER:

Illinois does not share a border with Kentucky, orIllinois is west of Missouri is false because~r is false and s is false.

37.

SOLUTION:

is a disjunction. A disjunction is true if at leastone of the statements is true. r is Illinois shares a border with Kentucky, which istrue. q is Illinois borders the Atlantic Ocean, whichis false. Then is true because r is true.

ANSWER:

Illinois shares a border with Kentucky, or Illinoisborders the Atlantic Ocean. is true because ris true.

38.

SOLUTION:

Negate both p and r and find the conjunction . A conjunction is true only when both

statements that form it are true. ~p is Springfield is not the capital of Illinois, whichis false. ~r is Illinois does not share a border withKentucky, which is false. Then is falsebecause ~p is false and ~r is false.

ANSWER:

Springfield is not the capital of Illinois, and Illinoisdoes not share a border with Kentucky. is false because ~p is false and ~r is false.

39.

SOLUTION:

Negate both s and p and find the disjunction. A disjunction is true if at least one of the

statements is true. ~s is Illinois is not west of Missouri, which is true.~p is Springfield is not the capital of Illinois, whichis false. Then is true because ~s is true.

ANSWER:

Illinois is not west of Missouri, or Springfield is notthe capital of Illinois. is true because ~s istrue.



Write each statement in if-then form.40. Get a free milkshake with any combo purchase.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If you buy a combo, then you get a free milkshake.

ANSWER:

If you buy a combo, then you get a free milkshake.

41. Everybody at the party received a gift.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If you were at the party, then you received a gift.

ANSWER:

If you were at the party, then you received a gift.

42. The intersection of two planes is a line.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If two planes intersect, then the intersection is aline.

ANSWER:

If two planes intersect, then the intersection is aline.

43. The area of a circle is

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If a figure is a circle, then the area is .

ANSWER:

If a figure is a circle, then the area is .

44. Collinear points lie on the same line.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If points are collinear, then they lie on the sameline.

ANSWER:

If points are collinear, then they lie on the same line.

45. A right angle measures 90 degrees.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If an angle is right, then the angle measures 90degrees.

ANSWER:

If an angle is right, then the angle measures 90degrees.



CONSTRUCT ARGUMENTS Determine thetruth value of each conditional statement. Iftrue, explain your reasoning. If false, give acounterexample.

46. If a banana is blue, then an apple is a vegetable.

SOLUTION:

The conditional statement "If a banana is blue, thenan apple is a vegetable." is true. The hypothesis isfalse, since a banana is never blue. A conditionalwith a false hypothesis is always true, so thisconditional statement is true.

ANSWER:

True; the hypothesis is false, since a banana is neverblue. A conditional with a false hypothesis is alwaystrue, so this conditional statement is true.

47. If a number is odd, then it is divisible by 5.

SOLUTION:

False; 9 is an odd number, but not divisible by 5. Thehypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

ANSWER:

False; 9 is an odd number, but not divisible by 5. Thehypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

48. If a dog is an amphibian, then the season is summer.

SOLUTION:

The conditional statement "If a dog is an amphibian,then the season is summer." is true. The hypothesisis false, since a dog is not an amphibian. Aconditional with a false hypothesis is always true, sothis conditional statement is true.

ANSWER:

True; the hypothesis is false, since a dog is not anamphibian. A conditional with a false hypothesis isalways true, so this conditional statement is true.

49. If an angle is acute, then it has a measure of 45.

SOLUTION:

False; the angle drawn is an acute angle whosemeasure is not 45. The hypothesis of the conditionalis true, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false.

ANSWER:

False; the angle drawn is an acute angle whosemeasure is not 45. The hypothesis of the conditionalis true, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false.

50. If a polygon has six sides, then it is a regularpolygon.

SOLUTION:

False; this polygon has six sides, but is not regular.The hypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

ANSWER:

False; this polygon has six sides, but is not regular.The hypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.



CONSTRUCT ARGUMENTS Determine thetruth value of each conditional statement. Iftrue, explain your reasoning. If false, give acounterexample.

51. If an angle’s measure is 25, then the measure of theangle’s complement is 65.

SOLUTION:

The conditional statement "If an angle’s measure is25, then the measure of the angle’s complement is65." is true. When this hypothesis is true, theconclusion is also true, since an angle and itscomplement’s sum is 90. So, the conditionalstatement is true.

ANSWER:

True; when this hypothesis is true, the conclusion isalso true, since an angle and its complement’s sumis 90. So, the conditional statement is true.

52. If North Carolina is south of Florida, then the capitalof Ohio is Columbus.

SOLUTION:

The conditional statement "If North Carolina is southof Florida, then the capital of Ohio is Columbus." istrue. The hypothesis is false, since North Carolina isnot south of Florida. A conditional with a falsehypothesis is always true, so this conditionalstatement is true.

ANSWER:

True; the hypothesis is false, since North Carolina isnot south of Florida. A conditional with a falsehypothesis is always true, so this conditionalstatement is true.

53. If red paint and blue paint mixed together makewhite paint, then 3 – 2 = 0.

SOLUTION:

The conditional statement "If red paint and bluepaint mixed together make white paint, then 3 – 2 =0" is true. The hypothesis is false, since red and bluepaint make green paint. A conditional with a falsehypothesis is always true, so this conditionalstatement is true.

ANSWER:

True; the hypothesis is false, since red and bluepaint make green paint. A conditional with a falsehypothesis is always true, so this conditionalstatement is true.

54. If two angles are congruent, then they are verticalangles.

SOLUTION:

False; the angles are congruent, but they are notvertical angles. The hypothesis of the conditional istrue, but the conclusion is false.This counterexample shows that the conditionalstatement is false.

ANSWER:

False; the angles are congruent, but they are notvertical angles. The hypothesis of the conditional istrue, but the conclusion is false. Thiscounterexample shows that the conditionalstatement is false.



55. If an animal is a bird, then it is an eagle.

SOLUTION:

The statement "If an animal is a bird, then it is aneagle." is false. The animal could be a falcon. Thehypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

ANSWER:

False; the animal could be a falcon. The hypothesisof the conditional is true, but the conclusion is false.This counterexample shows that the conditionalstatement is false.

56. If two angles are acute, then they aresupplementary.

SOLUTION:

False; and are acute, but their sum is 90°.The hypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

ANSWER:

False; and are acute, but their sum is 90°.The hypothesis of the conditional is true, but theconclusion is false. This counterexample shows thatthe conditional statement is false.

57. If two lines intersect, then they form right angles.

SOLUTION:

False; These lines intersect, but do not form rightangles. The hypothesis of the conditional is true, butthe conclusion is false. This counterexample showsthat the conditional statement is false.

ANSWER:

False; these lines intersect, but do not form rightangles. The hypothesis of the conditional is true, butthe conclusion is false. This counterexample showsthat the conditional statement is false.



Write the converse, inverse, andcontrapositive of each true conditionalstatement.Determine whether each related conditional istrue or false. If a statement is false, find acounterexample.

58. A right triangle has an angle measure of 90.

SOLUTION:

If a triangle is right, then it has an angle measure of90. The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If a triangle has an angle measure of 90,then it is a right triangle. The converse is true. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If a triangle is not right, then it does nothave an angle measure of 90. The inverse is true. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If a triangle does not have an anglemeasure of 90, then it is not a right triangle. Thecontrapositive is true.

ANSWER:

If a triangle is right, then it has an angle measure of90. Converse: If a triangle has an angle measure of90, then it is a right triangle; true. Inverse: If atriangle is not right, then it does not have an anglemeasure of 90; true. Contrapositive: If a triangledoes not have an angle measure of 90, then it is nota right triangle; true.

59. If you live in Chicago, you live in Illinois.

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If you live in Illinois, then you live inChicago. The converse is false.Counterexample: You can live in Urbana. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If you do not live in Chicago, then you donot live in Illinois. The inverse is false.Counterexample: You can live in Urbana. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If you do not live in Illinois, then youdo not live in Chicago. The contrapositive is true.

ANSWER:

Converse: If you live in Illinois, then you live inChicago. False: You can live in Galveston. Inverse:If you do not live in Chicago, then you do not live inIllinois. False: You can live in Galveston.Contrapositive: If you do not live in Illinois, then youdo not live in Chicago; true.



60. If a bird is an ostrich, then it cannot fly.

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If a bird cannot fly, then it is an ostrich.The converse is false.Counterexample: The bird could be a penguin. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If a bird is not an ostrich, then it can fly.The inverse is false.Counterexample: The bird could be a penguin. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If a bird can fly, then the bird is notan ostrich; The contrapositive is true.

ANSWER:

Converse: If a bird cannot fly, then it is an ostrich.False; The bird could be a penguin. Inverse: If a birdis not an ostrich, then it can fly. False; The birdcould be a penguin. Contrapositive: If a bird can fly,then the bird is not an ostrich; true.

61. If two angles have the same measure, then theangles are congruent.

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If two angles are congruent, then theyhave the same measure. The converse is true. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If two angles do not have the samemeasure, then the angles are not congruent. Theinverse is true. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If two angles are not congruent, thenthey do not have the same measure. Thecontrapositive is true.

ANSWER:

Converse: If two angles are congruent, then theyhave the same measure; true. Inverse: If two anglesdo not have the same measure, then the angles arenot congruent; true. Contrapositive: If two anglesare not congruent, then they do not have the samemeasure; true.



62. All congruent segments have the same length.

SOLUTION:

If segments are congruent, then they have the samelength. The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If segments have the same length, thenthey are congruent. The converse is true. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If segments are not congruent, then they donot have the same length. The inverse is true. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If segments do not have the samelength, then they are not congruent. Thecontrapositive is true.

ANSWER:

If segments are congruent, then they have the samelength. Converse: If segments have the same length,then they are congruent; true. Inverse: If segmentsare not congruent, then they do not have the samelength; true. Contrapositive: If segments do not havethe same length, then they are not congruent; true.

63. All squares are rectangles.

SOLUTION:

If a figure is a square, then it is a rectangle. The converse is formed by exchanging thehypothesis and conclusion of the conditional.Converse: If a figure is a rectangle, then it is asquare. The converse is false.Counterexample: A rectangle does not have to haveall sides congruent. The inverse is formed by negating both thehypothesis and conclusion of the conditional.Inverse: If a figure is not a square, then it is not arectangle. The inverse is false.Counterexample: The figure could be a rectangle,even though it is not a square. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.Contrapositive: If a figure is not a rectangle, then itis not a square. The contrapositive is true.

ANSWER:

If a figure is a square, then it is a rectangle.Converse: If a figure is a rectangle, then it is asquare. False. A rectangle does not have to have allsides congruent. Inverse: If a figure is not a square,then it is not a rectangle. False. The figure could bea rectangle, even though it is not a square.Contrapositive: If a figure is not a rectangle, then itis not a square; true.



Rewrite each statement as a biconditionalstatement. Then determine whether thebiconditional is true or false.

64. Lines that do not intersect are horizontal.

SOLUTION: Write a biconditional with if and only if. Lines do not intersect if and only if they arehorizontal; false they do not intersect if they areparallel or skew, they could in any orientation.

ANSWER: Lines do not intersect if and only if they arehorizontal; false.

65. Points that lie in the same plane are coplanar.

SOLUTION: Write a biconditional using if and only if. Points are coplanar if and only if they lie in the sameplane; true this is the definition of coplanar.

ANSWER: Points are coplanar if and only if they lie in the sameplane; true.

66. Right angles measure 90 degrees.

SOLUTION: Write a biconditional using if and only if. An angle is right if and only if it measures 90degrees; true this is the definition of a right angle.

ANSWER: An angle is right if and only if it measures 90degrees; true.

67. The midpoint of a segment bisects the segment.

SOLUTION: Write a biconditional using if and only if. A point is the midpoint of a segment if and only if itis on a line that bisects the segment; true a bisectoris a line through the midpoint.

ANSWER: A point is the midpoint of a segment if and only if itis on a line that bisects the segment; true.

68. Real numbers are irrational numbers.

SOLUTION: Write a biconditional using if and only if. A number is a real number if and only if it is anirrational number; false all irrational numbers arereal numbers, but all rational numbers are notirrational, but they are real.

ANSWER: A number is a real number if and only if it is anirrational number; false.

69. Perpendicular lines meet at right angles.

SOLUTION: Write as a biconditional using if and only if. Lines are perpendicular if and only if they meet atright angles; true as this is the definition ofperpendicular.

ANSWER: Lines are perpendicular if and only if they meet atright angles; true.



JUSTIFY ARGUMENTS Write the statementindicated, and use the information at the left todetermine the truth value of each statement. If astatement is false, give a counterexample. Animals with stripes are zebras.

70. conditional

SOLUTION:

To write these statements in conditional form,identify the hypothesis and conclusion. The word ifis not part of the hypothesis. The word then is notpart of the conclusion.If an animal has stripes, then it is a zebra. Theconditional is false. Counterexample: A tiger hasstripes.

ANSWER:

If an animal has stripes, then it is a zebra; false: atiger has stripes.

71. converse

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional.If an animal is a zebra, then it has stripes. Theconverse is true.

ANSWER:

If an animal is a zebra, then it has stripes; true.

72. inverse

SOLUTION:

The inverse is formed by negating both thehypothesis and conclusion of the conditional.If an animal does not have stripes, then it is not azebra. The inverse is true.

ANSWER:

If an animal does not have stripes, then it is not azebra; true.

73. contrapositive

SOLUTION:

The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.If an animal is not a zebra, then it does not havestripes. The contrapositive is false. Counterexample:A tiger has stripes.

ANSWER:

If an animal is not a zebra, then it does not havestripes; false: a tiger has stripes.

Write the conditional and converse for eachstatement. Determine the truth values for theconditionals and converses. If false, write acounterexample. Write a biconditional ifpossible.

74. Regular quadrilaterals are squares.

SOLUTION: Write as if then.Conditional: If a quadrilateral is regular then it is asquare; true all sides and angles are congruent in asquare. Reverse the order of the conditional.Converse: If a quadrilateral is a square then it isregular; true all sides and angles are congruent in asquare. Write using if and only if.Biconditional: A quadrilateral is regular if and only ifit is a square; true all sides and angles are congruentin a square.

ANSWER: Conditional: If a quadrilateral is regular then it is asquare; true. Converse: If a quadrilateral is a square then it isregular; true. Biconditional: A quadrilateral is regular if and only ifit is a square; true.



75. Equilateral triangles have all sides the same length.

SOLUTION: Write as if then.Conditional: If a triangle is equilateral then all sideshave the same length; true this is definition ofequilateral. Reverse the parts of the if thenConverse: If a triangle has all sides the same length,then it is equilateral; true this is definition ofequilateral. Write with if and only if.Biconditional: A triangle is equilateral if and only ifall sides have the same length; true this is definitionof equilateral.

ANSWER: Conditional: If a triangle is equilateral then all sideshave the same length; true. Converse: If a triangle has all sides the same length,then it is equilateral; true. Biconditional: A triangle is equilateral if and only ifall sides have the same length; true.

76. Right angles have measures greater than acuteangles.

SOLUTION: Right angles have measures greater than acuteangles. Write as if then and test statement.Conditional: If an angle is right, then it measuresgreater than an acute angle; true. Reverse the statement. Converse: If an angle measures greater than anacute angle, then it is right; false an obtuse angle isalso greater than an acute angle.

ANSWER: Conditional: If an angle is right, then it measuresgreater than an acute angle; true. Converse: If an angle measures greater than anacute angle, then it is right; false an obtuse angle isalso greater than an acute angle.

77. Integers are rational numbers.

SOLUTION: Write the statement using if then.Conditional: If a number is an integer then it is arational number; true. All integers can be written as fraction with one asdenominator. Reverse the order of the conditional and test thenew statement. Converse: If a number is a rational number then it isan integer; false 0.5 is a rational number that is notan integer.

ANSWER: Conditional: If a number is an integer then it is arational number; true. Converse: If a number is a rational number then it isan integer; false 0.5 is a rational number that is notan integer.



78. VEHICLES Different vehicles are characterizedby different structural features. Write eachstatement in if-then form.• A convertible describes any vehicle with a fullyretractable top.• A coupe is any car with only two full-sizepassenger doors.• A pickup is any vehicle with an open cargo bed inthe rear.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion. If you drive a convertible, then you have a vehiclewith a fully retractable top. If you drive a coupe, then you have a car with onlytwo full-size passenger doors. If you drive a pickup, then you drive a vehicle withan open cargo bed in the rear.

ANSWER:

If you drive a convertible, then you have a vehiclewith a fully retractable top. If you drive a coupe, then you have a car with onlytwo full-size passenger doors. If you drive a pickup, then you drive a vehicle withan open cargo bed in the rear.

79. ART Write the following statement in if-then form:At the Andy Warhol Museum in Pittsburgh,Pennsylvania, most of the collection is AndyWarhol's artwork.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If the museum is the Andy Warhol Museum, thenmost of the collection is Andy Warhol's artwork.

ANSWER:

If the museum is the Andy Warhol Museum, thenmost of the collection is Andy Warhol's artwork.



80. SCIENCE The water on Earth is constantlychanging through a process called the water cycle.Write the three conditionals below in if-then form.

a. As runoff, water flows into bodies of water.b. Plants return water to the air throughtranspiration.c. Water bodies return water to the air throughevaporation.

SOLUTION:

To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.a. If water runs off, it flows into bodies of water.b. If plants return water to the air, they transpire.c. If water bodies return water to the air, it isthrough evaporation.

ANSWER:

a. If water runs off, it flows into bodies of water.b. If plants return water to the air, they transpire.c. If water bodies return water to the air, it isthrough evaporation.

81. SPORTS In football, touchdowns are worth 6points, extra point conversions are worth 2 points,and safeties are worth 2 points.a. Write three conditional statements in if-then formfor scoring in football.b. Write the converse of the three true conditionalstatements. State whether each is true or false. If astatement is false, find a counterexample.

SOLUTION:

a. To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If a football team makes a touchdown, they get 6points; If a football team makes a two-pointconversion, they get 2 points; If a football teammakes a safety, they get 2 points.b. The converse is formed by exchanging thehypothesis and conclusion of the conditional.If a football team gets 6 points, they made atouchdown. The converse is true. If a football team gets 2 points, they made a two-point conversion. The converse is false.Counterexample: They could have gotten a safety;If a football team gets 2 points, they made a safety.the converse is false. Counterexample, They couldhave gotten a two-point conversion.

ANSWER:

a. Sample answer: If a football team makes atouchdown, they get 6 points; If a football teammakes a two-point conversion, they get 2 points; If afootball team makes a safety, they get 2 points.b. Sample answer: If a football team gets 6 points,they made a touchdown. True; If a football teamgets 2 points, they made a two-point conversion.False; they could have gotten a safety; If a footballteam gets 2 points, they made a safety. False; theycould have gotten a two-point conversion.

82. SCIENCE Chemical compounds are grouped anddescribed by the elements that they contain. Acidscontain hydrogen (H). Bases contain hydroxide(OH). Hydrocarbons contain only hydrogen (H) and



carbon (C).

a. Write three conditional statements in if-then formfor classifying chemical compounds.b. Write the converse of the three true conditionalstatements.

SOLUTION:

a. To write these statements in if-then form, identifythe hypothesis and conclusion. The word if is notpart of the hypothesis. The word then is not part ofthe conclusion.If a compound is an acid, it contains hydrogen. If acompound is a base, it contains hydroxide. If acompound is a hydrocarbon, it contains onlyhydrogen and carbon.b. The converse is formed by exchanging thehypothesis and conclusion of the conditional.If a compound contains hydrogen, it is an acid. Theconverse is false. Counterexample: A hydrocarboncontains hydrogen. If a compound contains hydroxide, it is a base. Theconverse is true. If a compound contains only hydrogen and carbon, itis a hydrocarbon. The converse is true.

ANSWER:

a. Sample answer: If a compound is an acid, itcontains hydrogen.If a compound is a base, it contains hydroxide. If acompound is a hydrocarbon, it contains onlyhydrogen and carbon.b. Sample answer: If a compound containshydrogen, it is an acid.False; a hydrocarbon contains hydrogen. If acompound contains hydroxide, it is a base; true. If acompound contains only hydrogen and carbon, it is ahydrocarbon; true.

83. Can the conditional "If , then x = 6," becombined with its converse to form a truebiconditional? Explain.

SOLUTION: No, the biconditional cannot be formed, because theconditional is not true. The value of x could also be-6.

ANSWER: No, the biconditional cannot be formed, because theconditional is not true. The value of x could also be-6.

84. If the contrapositive of a conditional is true, can yourewrite the conditional as a true biconditional?Explain.

SOLUTION: No, the contrapositive has the same truth as theconditional, which means the conditional is true, butthe converse must also be true, but that informationis not given.

ANSWER: No, the contrapositive has the same truth as theconditional, which means the conditional is true, butthe converse must also be true, but that informationis not given.

85. Compare the mathematical meanings of the symbols and in and .

SOLUTION: The expression with the right arrow means theconditional "if p is true then q is also true." The expression with the double arrow means thereis a conditional "p is true if and only if q is true."The arrow tells which way(s) the conditional goes.

ANSWER: The expression with the right arrow means theconditional "if p is true then q is also true." The expression with the double arrow means thereis a conditional "p is true if and only if q is true."The arrow tells which way(s) the conditional goes.



86. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate a law of logic by usingconditionals.a. LOGICAL Write three true conditionalstatements, using each consecutive conclusion asthe hypothesis for the next statement.b. LOGICAL Write a conditional using thehypothesis of your first conditional and theconclusion of your third conditional. Is theconditional true if the hypothesis is true?c. VERBAL Given two conditionals If a, then band If b, then c, make a conjecture about the truthvalue of c when a is true. Explain your reasoning.

SOLUTION:

a.With consecutive conclusion, the conclusion is thehypothesis of the next statement. If a, then b. If b,then c, If c, then d. If you live in New York City, then you live in NewYork State; If you live in New York State, then youlive in the United States; If you live in the UnitedStates, then you live in North America.b. The hypothesis of the first statement is "If youlive in New York City". The conclusion of the thirdstatement is " you live in North America". Then thenew conditional is "If you live in New York City,then you live in North America" . The newconditional statement is true.c. If a is true, then c is true. If we know that a istrue, then we know that b is true, and if we knowthat b is true, then we know that c is true.Therefore, when a is true, c is true.

ANSWER:

a. Sample answer: If you live in New York City,then you live in New York State; If you live in NewYork State, then you live in the United States; If youlive in the United States, then you live in NorthAmerica.b. If you live in New York City, then you live inNorth America; yes.c. Sample answer: If a is true, then c is true. If weknow that a is true, then we know that b is true,and if we know that b is true, then we know that cis true. Therefore, when a is true, c is true.

LOGIC To negate a statement containing thewords all or for every, you can use the phrase atleast one or there exists. To negate a statementcontaining the phase there exists, you can use thephrase 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. Forexample, The square of a real number isnonnegative implies the following conditional and itsnegation.

p: For every real number x, x2 ≥ 0.

~p: There exists a real number x such that x2 < 0 . Use the information given to write the negationof each statement.

87. There exists a segment that has no midpoint.

SOLUTION:

To negate a statement containing the phrase "thereexists", use the word "every".Every segment has a midpoint.

ANSWER:

Every segment has a midpoint.

88. Every student at Hammond High School has alocker.

SOLUTION:

To negate a statement containing the word "every",use the phase "at least one".There exists at least one student at Hammond HighSchool who does not have a locker.

ANSWER:

There exists at least one student at Hammond HighSchool who does not have a locker.



89. All squares are rectangles.

SOLUTION:

To negate a statement containing the word "all", usethe phase "at least one".There exists at least one square that is not arectangle.

ANSWER:

There exists at least one square that is not arectangle.

90. There exists a real number x such that

SOLUTION:

To negate a statement containing the phrase "thereexists", use the word "every".

For every real number x, x2 .

ANSWER:

For every real number x, x2 .

91. There exists an even number x such that .

SOLUTION:

To negate a statement containing the phrase "thereexists", use the phrase "for every".For every even number x, .

ANSWER:

For every even number x, .

92. Every real number has a real square root.

SOLUTION:

To negate a statement containing the word "every",use the phrase "there exists" .There exists a real number that does not have a realsquare root.

ANSWER:

There exists a real number that does not have a realsquare root.

93. ERROR ANALYSIS Nicole and Kiri areevaluating the conditional If 15 is a prime number,then 20 is divisible by 4. Both think that theconditional is true, but their reasoning differs. Iseither of them correct? Explain.

SOLUTION:

Kiri is correct. For the conditional statement "If 15is a prime number, then 20 is divisible by 4." p is 15is "a prime number", which is false and q is "20 isdivisible by 4" is true. When the hypothesis of aconditional is false, the conditional is always true.Nicole, did not consider the truth value of thehypothesis.

ANSWER:

Sample answer: Kiri; when the hypothesis of aconditional is false, the conditional is always true.



94. CHALLENGE You have learned that statementswith the same truth value are logically equivalent.Use logical equivalence to summarize theconditional, converse, inverse, and contrapositive forthe statements p and q.

SOLUTION:

The conditional statement is (p → q). The converseis formed by exchanging the hypothesis andconclusion of the conditional (q → p). The inverse isformed by negating both the hypothesis andconclusion of the conditional (~p → ~q). Thecontrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional (~q → ~p). If the conditional is true, then the contrapositive isalso true. These are logically equivalent. Similarly, if the converse is true, then the inverse isalso true. These are also logically equivalent.

ANSWER:

If the conditional is true, then the contrapositive isalso true. These are logically equivalent. Similarly, if the converse is true, then the inverse isalso true. These are also logically equivalent.

95. REASONING You are evaluating a conditionalstatement in which the hypothesis is true, but theconclusion is false. Is the inverse of the statementtrue or false? Explain your reasoning.

SOLUTION:

The inverse of a conditional statement in which thehypothesis is true is true. Since the conclusion isfalse, theconverse of the statement must be true. Theconverse and inverse are logically equivalent, so theinverse is also true.Consider the following truth table.

Conditional Converse Inversep q p→q q→p ~p→~qT F F T T

ANSWER:

True; since the conclusion is false, the converse ofthe statement must be true. The converse andinverse are logically equivalent, so the inverse is alsotrue.



96. OPEN ENDED Write a conditional statement inwhich the converse, inverse, and contrapositive areall true. Explain your reasoning.

SOLUTION:

If four is divisible by two, then birds have feathers.In order for the converse, inverse, andcontrapositive to be true, the hypothesis and theconclusion must both be either true or false. p = " four is divisible by two" and q = "birds havefeathers"

ANSWER:

Sample answer: If four is divisible by two, then birdshave feathers. In order for the converse, inverse,and contrapositive to be true, the hypothesis and theconclusion must both be either true or false.

97. CHALLENGE The inverse of conditional A isgiven below. Write conditional A, its converse, andits contrapositve. Explain your reasoning.If I received a detention, then I did not arrive atschool on time.

SOLUTION:

The inverse is formed by negating both thehypothesis and conclusion of the conditional.The hypothesis q of the inverse statement is Ireceived a detention. The conclusion p of theinverse statement is I did not arrive at school ontime. So the conditional A is : If I did not arrive atschool on time, then I received a detention. The converse is formed by exchanging thehypothesis and conclusion of the conditional.So the converse of statement A is : If Iarrived at school on time, then I did not receive adetention. The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional.The contrapositive of Statement A is : IfI did not receive a detention, then I arrived at schoolon time.

ANSWER:

The hypothesis q of the inverse statement is Ireceived a detention. The conclusion p of theinverse statement is I did not arrive at school ontime. So the conditional A is : If I did notarrive at school on time, then I received a detention.So the converse of statement A is : If Iarrived at school on time, then I did not receive adetention. The contrapositive of Statement A is

: If I did not receive a detention, then Iarrived at school on time.



98. WRITING IN MATH Describe the relationshipbetween a conditional, its converse, its inverse, andits contrapositive.

SOLUTION:

The converse is formed by exchanging thehypothesis and conclusion of the conditional. Theinverse is formed by negating both the hypothesisand conclusion of the conditional.The contrapositiveis formed by negating both the hypothesis and theconclusion of the converse of the conditional.Since they are logically equivalent, a conditional andits contrapositive always have the same truth value.The inverse and converse of a conditional are alsologically equivalent and have the same truth value.The conditional and its contrapositive can have thesame truth value as its inverse and converse, or itcan have the opposite truth value of its inverse andconverse.

ANSWER:

Sample answer: Since they are logically equivalent,a conditional and its contrapositive always have thesame truth value. The inverse and converse of aconditional are also logically equivalent and have thesame truth value. The conditional and itscontrapositive can have the same truth value as itsinverse and converse, or it can have the oppositetruth value of its inverse and converse.

99. WRITING IN MATH If a biconditional is true,what do you know about the conditional and theconverse? If the biconditional is false, what do youknow about the conditional and converse?

SOLUTION: If a biconditional is true, then the conditional and theconverse are both true. If a biconditional is false,then the conditional, the converse, or both are false.

ANSWER: If a biconditional is true, then the conditional and theconverse are both true. If a biconditional is false,then the conditional, the converse, or both are false.

100. REASONING Because a conditional statementand its converse must both be true for abiconditional to be true, the order of the hypothesisand the conclusion do not matter. How does thisaffect the truth values of the statements? Explain.

SOLUTION: Since the conditional and its converse are both true,interchanging the order of the hypothesis andconclusion does not affect the truth values of thestatements.

ANSWER: Since the conditional and its converse are both true,interchanging the order of the hypothesis andconclusion does not affect the truth values of thestatements.

101. MULTI-STEP Use conditional statements Ithrough IV to answer the following questions. I. Two lines are perpendicular, and the linesintersect.II. All triangles have an acute angle. III. The sum of the measures of twosupplementary angles is 180 degrees.IV. An angle is a right angle, and it has ameasurement of 90 degrees. a. Rewrite each of the conditional statements in if-then form. Then, write the converse statement. b. Which conditional statements have a trueconverse? A I B II C III D IV c. Which statements, paired with their conversescan be written as a biconditional? Select all of thetrue statements. A I B II C III D IV

SOLUTION:



a. Write each first as an if then, then reverse theorder. I. If two lines are perpendicular then the linesintersect. Converse: If two lines intersect, then theyare perpendicular.II. If a figure is a triangle then it has an acute angle.Converse: If a figure has an acute angle, then it is atriangle. III. If the sum of the measures of two angles is 180degrees, then the angles are supplementary.Converse: If two angles are supplementary, then thesum of their measures is 180 degrees.IV. If an angle is a right angle, then the anglemeasures 90 degrees. Converse: If an angle measures 90 degrees, then the angle is a rightangle. b. C, D these are the definitions, but A is not true,because lines can intersect and not be perpendicular,and B is not true, because quadrilaterals and otherpolygons can also have acute angles. c. C, D see explanation in part b.

ANSWER: a. I. If two lines are perpendicular then the linesintersect. Converse: If two lines intersect, then theyare perpendicular.II. If a figure is a triangle then it has an acute angle.Converse: If a figure has an acute angle, then it is atriangle. III. If the sum of the measures of two angles is 180degrees, then the angles are supplementary.Converse: If two angles are supplementary, then thesum of their measures is 180 degrees.IV. If an angle is a right angle, then the anglemeasures 90 degrees. Converse: If an angle measures 90 degrees, then the angle is a rightangle. b. C, Dc. C, D

102. Which of the following can be used to prove that aconditional statement is false? A counterexample B converse C conclusion D contrapositive

SOLUTION: A counterexample can be used to prove aconditional statement is false by finding an examplewhere the hypothesis is true and the conclusion isfalse.

ANSWER: A



103. Consider the conditional statements below.

I. Every eagle is a bird.II. A butterfly is an insect.III. If an animal is a tiger, then it lives underwater.

Which of the conditional statements has a truecontrapositive?A I onlyB II onlyC III onlyD I and II onlyE I, II, and III

SOLUTION: A conditional and its contrapositive are logicallyequivalent. Logically equivalent statements have thesame truth value. To identify the conditionalstatements that have a true contrapositive, identifythe true conditional statements. I. Every eagle is a bird. The conditional statement is If a creature is aneagle, then it is a bird. This conditional statementis true, so the contrapositive is also true. II. A butterfly is an insect.The conditional statement is If a creature is abutterfly, then it is an insect. This conditionalstatement is true, so the contrapositive is also true. III. If an animal is a tiger, then it lives underwater.This conditional statement is false, so thecontrapositive is also false. Statements I and II have a true contrapositive.

ANSWER: D

104. Which of the following statements is logicallyequivalent to the statement below. Select all logicallyequivalent statements. All octagons are polygons. A If a figure is not a polygon, then it is not anoctagon. B All polygons are octagons. C If a figure is not an octagon, then it is nota polygon. D Every polygon is also an octagon. E A figure is an octagon if and only if it is apolygon.

SOLUTION: Compare the statements. All octagons are polygons. A If a figure is not a polygon, then it is not anoctagon. = Contrapositive B All polygons are octagons. = Converse C If a figure is not an octagon, then it is nota polygon. = InverseD Every polygon is also an octagon. = ConverseE A figure is an octagon if and only if it is a polygon= Biconditional The converse is false, because octagons are onlypolygons with 8 sides, so only A is true.

ANSWER: A



105. Use the following statements to write a compoundstatement. Then find the truth value. p: A triangle has two congruent sides.q: A triangle has no congruent sides.r: A triangle is equilateral. a. b.

SOLUTION: p: A triangle has two congruent sides.q: A triangle has no congruent sides.r: A triangle is equilateral. a. = A triangle has two congruent sides,no congruent sides, or is equilateral. True, becausethere are three sides in a triangle. b. = A triangle has two congruent sides and nocongruent sides, or is equilateral. False, a trianglecannot have both 0 and 2 sides congruent.

ANSWER: a. A triangle has two congruent sides, no congruentsides, or is equilateral; true. b. A triangle has two congruent sides and nocongruent sides, or is equilateral; false.



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2-2 Statements, Conditionals, and Biconditionals