2-2 Statements, Conditionals, and Biconditionals Use the following statements to write a 3. compound statement for each conjunction or SOLUTION: disjunction. Then find its truth value. Explain your reasoning. is a disjunction. A disjunction is true if at least p: A week has seven days. one of the statements is true. q is false, since there q: There are 20 hours in a day. are 24 hours in a day, not 20 hours. r is true since r: There are 60 minutes in an hour. there are 60 minutes in an hour. Thus, is true, 1. p and r because r is true. SOLUTION: ANSWER: p and r is a conjunction. A conjunction is true only There are 20 hours in a day, or there are 60 minutes when both statements that form it are true. p is true in an hour. since a week has seven day. r is true since there are is true, because r is true. 60 minutes in an hour. Then p and r is true, 4. because both p and r are true. SOLUTION: ANSWER: ~p is a negation of statement p, or the opposite of A week has seven days, and there are 60 minutes in statement p. The or in ~p or q indicates a an hour. p and r is true, because p is true and r is disjunction. A disjunction is true if at least one of the true. statements is true. ~p would be : A week does not have seven days, 2. which is false. q is false since there are 24 hours in SOLUTION: a day, not 20 hours in a day. Then ~p or q is false, is a conjunction. A conjunction is true only because both ~p and q are false. when both statements that form it are true. p is true ANSWER: since a week has seven days. q is false since there A week does not have seven days, or there are 20 are 24 hours in a day, not 20 hours in a day. Thus, hours in a day. ~p or q is false, because ~p is false is false, because q is false. and q is false. ANSWER: 5. A week has seven days, and there are 20 hours in a day. is false, because q is false. SOLUTION: is a disjunction. A disjunction is true if at least one 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, is true, because p is true and r is true. ANSWER: A week has seven days, or there are 60 minutes in an hour. is true, because p is true and r is true. eSolutions Manual - Powered by Cognero Page 1 2-2 Statements, Conditionals, and Biconditionals 6. 9. The measure of an acute angle is between 0 and 90. SOLUTION: SOLUTION: ~p and ~r is the conjunction of the negations of p To write these statements in if-then form, identify and r. A conjunction is true if both statements are the hypothesis and conclusion. The word if is not true. part of the hypothesis. The word then is not part of ~p is : A week does not have seven days, which is the conclusion. false. ~r is : There are not 60 minutes in an hour, If the angle is acute, then its measure is between 0 which is false. Then is false, because and 90. both ~p and ~r are false. ANSWER: ANSWER: If the angle is acute, then its measure is between 0 A week does not have seven days, and there are not and 90. 60 minutes in an hour. is false, because 10. Equilateral triangles are equiangular. ~p is false and ~r is false. SOLUTION: Write each statement in if-then form. To write these statements in if-then form, identify 7. Sixteen-year-olds are eligible to drive. the hypothesis and conclusion. The word if is not SOLUTION: part of the hypothesis. The word then is not part of To write these statements in if-then form, identify the conclusion. the hypothesis and conclusion. The word if is not If a triangle is equilateral, then it is equiangular. part of the hypothesis. ANSWER: The word then is not part of the conclusion. If a triangle is equilateral, then it is equiangular. If you are sixteen years old, then you are eligible to Determine the truth value of each conditional drive. statement. If true, explain your reasoning. If false, give a counterexample. ANSWER: If you are sixteen years old, then you are eligible to 11. If then x = 4. drive. SOLUTION: If (–4)2 = 16. The hypothesis of the 8. Cheese contains calcium. conditional is true, but the conclusion is false. This SOLUTION: counterexample shows that the conditional To write these statements in if-then form, identify statement is false. the hypothesis and conclusion. The word if is not ANSWER: part of the hypothesis. The word then is not part of False; if x = –4, (–4)2 =16. The hypothesis of the the conclusion. conditional is true, but the conclusion is false. This If it is cheese, then it contains calcium. counterexample shows that the conditional ANSWER: statement is false If it is cheese, then it contains calcium. eSolutions Manual - Powered by Cognero Page 2 2-2 Statements, Conditionals, and Biconditionals 12. If you live in Atlanta, then you live in Georgia. 15. If the measure of a right angle is 95, then bees are lizards. SOLUTION: The conditional is false. You could live in Atlanta, SOLUTION: Georgia or Atlanta, Kansas. The conditional statement "If the measure of a right angle is 95, then bees are lizards." is true. The ANSWER: hypothesis is false, since the measure of a right False; Atlanta, Kansas; The hypothesis of the angle is 90. A conditional with a false hypothesis is conditional is true, but the conclusion is false. This always true, so this conditional statement is true. counterexample shows that the conditional statement is false. ANSWER: True; the hypothesis is false, since the measure of a 13. If tomorrow is Friday, then today is Thursday. right angle is 90. A conditional with a false SOLUTION: hypothesis is always true, so this conditional The conditional statement "If tomorrow is Friday, statement is true. then today is Thursday." is true. When this 16. If pigs can fly, then 2 + 5 = 7. hypothesis is true, the conclusion is also true, since Friday is the day that follows Thursday. So, the SOLUTION: conditional statement is true. The conditional statement "If pigs can fly, then 2 + 5 = 7" is true. The hypothesis is false, since pigs ANSWER: cannot fly. A conditional with a false hypothesis is True; when this hypothesis is true, the conclusion is always true, so this conditional statement is true. also true, since Friday is the day that follows Thursday. So, the conditional statement is true. ANSWER: True; the hypothesis is false, since pigs cannot fly. A 14. If an animal is spotted, then it is a Dalmatian. conditional with a false hypothesis is always true, so SOLUTION: this conditional statement is true. False 17. If Jacqueline turned 14 years old last year, then she The animal could be a leopard. The hypothesis of will turn 15 this year. the conditional is true, but the conclusion is false. This counterexample shows that the conditional SOLUTION: statement is false. Hypothesis: Jacqueline turned 14 years old last year Conclusion: She will turn 15 this year ANSWER: False; the animal could be a leopard. The hypothesis 14 + 1 = 15, so if the hypothesis is true, then the conclusion is true, so the conditional is true. of the conditional is true, but the conclusion is false. This counterexample shows that the conditional ANSWER: statement is false. True; the number 15 is one more than 14. eSolutions Manual - Powered by Cognero Page 3 2-2 Statements, Conditionals, and Biconditionals 18. If a number is between 10 and 12, then it is 11. 20. All whole numbers are integers. SOLUTION: SOLUTION: Consider all numbers between 10 and 12, 11 is If a number is a whole number, then it is an integer. between them, but so are 10.5 and 11.5 and many others. The converse is formed by exchanging the ANSWER: hypothesis and conclusion of the conditional. False; 11.5 is between 10 and 12. Converse: If a number is an integer, then it is a whole number. False; Sample answer: –3. JUSTIFY ARGUMENTS Write the converse, inverse, and contrapositive of each true The inverse is formed by negating both the conditional statement. hypothesis and conclusion of the conditional. Determine whether each related conditional is Inverse: If a number is not a whole number, then it true or false. If a statement is false, find a is not an integer. False: Sample answer: –3. counterexample. 19. If a number is divisible by 4, then it is divisible by 2. The contrapositive is formed by negating both the SOLUTION: hypothesis and the conclusion of the converse of the conditional. The converse is formed by exchanging the Contrapositive: If a number is not an integer, then it hypothesis and conclusion of the conditional. is not a whole number; true. Converse: If a number is divisible by 2, then it is divisible by 4; false. Sample answer: 6 is divisible by ANSWER: 2 but is not divisible by 4. If a number is a whole number, then it is an integer. Converse: If a number is an integer, then it is a The inverse is formed by negating both the whole number.