Question

directions: determine whether the conclusion follows from the given statements by the law of detachment of the law of syllogism. if it does, state which law was used. if not, write invalid. 8. given: if you go camping, then you will need a flashlight. given: if you need a flashlight, then you will need extra batteries. conclusion: if you go camping, then you will need extra batteries. answer: 9. given: if sally goes to the airport, then she will need to pay for parking. given: if sally does not pay for parking, then her car will get towed. conclusion: if sally does not go to the airport, then her car will not get towed. answer: 10. given: if you leave the country, then you will need a passport. given: nate is planning a trip to italy. conclusion: nate will need a passport. answer: 11. given: if you buy one pair of shoes, then you get another pair for 50% off. given: carolyn does not buy one pair of shoes. conclusion: carolyn does not get another pair for 50% off. answer: 12. given: if the sum of the measures of two angles is 180°, then they are supplementary. given: m∠d = 110° and m∠e = 70° conclusion: ∠d and ∠e are supplementary angles. answer: 13. given: if a number is a multiple of 16, then it is a multiple of 8. given: if a number is a multiple of 8, then it is a multiple of 4. conclusion: if a number is a multiple of 16, then it is a multiple of 4. answer:

Explanation:

Step1: Recall the Law of Detachment and Law of Syllogism

The Law of Detachment states if \(p
ightarrow q\) is true and \(p\) is true, then \(q\) is true. The Law of Syllogism states if \(p
ightarrow q\) and \(q
ightarrow r\) are true, then \(p
ightarrow r\) is true.

Step2: Analyze question 8

Let \(p =\) "you go camping", \(q=\) "you will need a flashlight", \(r = \) "you will need extra - batteries". We have \(p
ightarrow q\) and \(q
ightarrow r\). By the Law of Syllogism, \(p
ightarrow r\). So the conclusion follows by the Law of Syllogism.

Step3: Analyze question 9

Let \(p=\) "Sally goes to the airport", \(q = \) "Sally will need to pay for parking", \(r=\) "her car will get towed". We have \(p
ightarrow q\) and \(
eg q
ightarrow r\). The conclusion \(
eg p
ightarrow
eg r\) does not follow from the given statements. So the answer is invalid.

Step4: Analyze question 10

Let \(p=\) "you leave the country", \(q=\) "you will need a passport". Since Nate is planning a trip to Italy (leaving the country), \(p\) is true for Nate. Given \(p
ightarrow q\), by the Law of Detachment, \(q\) is true for Nate. So the conclusion follows by the Law of Detachment.

Step5: Analyze question 11

Let \(p=\) "you buy one pair of shoes", \(q=\) "you get another pair for 50% off". We have \(p
ightarrow q\) and \(
eg p\). The Law of Detachment requires \(p\) to be true to conclude \(q\). Here, just because \(
eg p\) is true, we cannot conclude \(
eg q\). So the answer is invalid.

Step6: Analyze question 12

Let \(p=\) "the sum of the measures of two angles is \(180^{\circ}\)", \(q=\) "they are supplementary". Given \(p
ightarrow q\) and \(m\angle D + m\angle E=110^{\circ}+70^{\circ}=180^{\circ}\) (so \(p\) is true), by the Law of Detachment, \(q\) is true. So the conclusion follows by the Law of Detachment.

Step7: Analyze question 13

Let \(p=\) "a number is a multiple of 16", \(q=\) "a number is a multiple of 8", \(r=\) "a number is a multiple of 4". We have \(p
ightarrow q\) and \(q
ightarrow r\). By the Law of Syllogism, \(p
ightarrow r\). So the conclusion follows by the Law of Syllogism.

Answer:

1. Law of Syllogism
2. invalid
3. Law of Detachment
4. invalid
5. Law of Detachment
6. Law of Syllogism