Question

name:
date:
per:
unit 2: logic & proof
homework 5: deductive reasoning
this is a 2 - page document
directions: use the law of detachment to give a valid conclusion. if not possible, write no valid conclusion.

1. given: if logan eats his vegetables, then he can have a bowl of ice - cream. logan eats his vegetables.

conclusion:

1. given: if a polynomial is prime, then it cannot be factored. 5x + 13y is prime.

conclusion:

1. given: if you visit paris, then you will see the eiffel tower. you did not see the eiffel tower.

conclusion:

1. given: if the measure of an angle is greater than 90°, then it is an obtuse angle. m∠pqr = 115°

conclusion:
directions: use the law of syllogism to give a valid conclusion. if not possible, write no valid conclusion.

1. given: if nicole is tardy to class again, she will get detention. if nicole gets detention, her mother will take away her phone.

conclusion:

1. given: if it is raining, then i will bring my umbrella. if i do not bring my umbrella, then i do not need my rain boots.

conclusion:

1. given: if a natural number ends in 0, then it is divisible by 10. if a natural number is divisible by 10, then it is divisible by 5.

conclusion:

Explanation:

Step1: Recall Law of Detachment

The Law of Detachment states that if \(p
ightarrow q\) is a true conditional statement and \(p\) is true, then \(q\) is true.

Step2: Solve problem 1

The conditional is "If Logan eats his vegetables, then he can have a bowl of ice - cream" (\(p
ightarrow q\)) and \(p\) (Logan eats his vegetables) is true. So, \(q\) is true.
Conclusion: Logan can have a bowl of ice - cream.

Step3: Solve problem 2

The conditional is "If a polynomial is prime, then it cannot be factored" (\(p
ightarrow q\)) and \(p\) (\(5x + 13y\) is prime) is true. So, \(q\) is true.
Conclusion: \(5x + 13y\) cannot be factored.

Step4: Solve problem 3

The conditional is "If you visit Paris, then you will see the Eiffel Tower" (\(p
ightarrow q\)). We know \(
eg q\) (you did not see the Eiffel Tower). By modus tollens (not part of Law of Detachment but related logical rule), \(
eg p\) (you did not visit Paris). But using only Law of Detachment, no valid conclusion.
Conclusion: no valid conclusion

Step5: Solve problem 4

The conditional is "If the measure of an angle is greater than \(90^{\circ}\), then it is an obtuse angle" (\(p
ightarrow q\)). Given \(m\angle PQR=115^{\circ}\) (\(p\) is true as \(115^{\circ}>90^{\circ}\)). So, \(q\) is true.
Conclusion: \(\angle PQR\) is an obtuse angle.

Step6: Recall Law of Syllogism

The Law of Syllogism states that if \(p
ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\).

Step7: Solve problem 5

The first conditional is "If Nicole is tardy to class again, she will get detention" (\(p
ightarrow q\)) and the second is "If Nicole gets detention, her mother will take away her phone" (\(q
ightarrow r\)).
Conclusion: If Nicole is tardy to class again, her mother will take away her phone.

Step8: Solve problem 6

The first conditional is "If it is raining, then I will bring my umbrella" (\(p
ightarrow q\)) and the second is "If I do not bring my umbrella, then I do not need my rain - boots" (\(
eg q
ightarrow
eg r\)). There is no way to form a valid chain for Law of Syllogism.
Conclusion: no valid conclusion

Step9: Solve problem 7

The first conditional is "If a natural number ends in 0, then it is divisible by 10" (\(p
ightarrow q\)) and the second is "If a natural number is divisible by 10, then it is divisible by 5" (\(q
ightarrow r\)).
Conclusion: If a natural number ends in 0, then it is divisible by 5.

Answer:

1. Logan can have a bowl of ice - cream.
2. \(5x + 13y\) cannot be factored.
3. no valid conclusion
4. \(\angle PQR\) is an obtuse angle.
5. If Nicole is tardy to class again, her mother will take away her phone.
6. no valid conclusion
7. If a natural number ends in 0, then it is divisible by 5.