Question

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∠pok = 119°

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 true and \(p\) is true, then \(q\) is true.

Step2: Solve problem 1

Given \(p
ightarrow q\): If Logan eats his vegetables, then he can have a bowl of ice - cream. And \(p\): Logan eats his vegetables. By Law of Detachment, \(q\) is true. So the conclusion is Logan can have a bowl of ice - cream.

Step3: Solve problem 2

Given \(p
ightarrow q\): If a polynomial is prime, then it cannot be factored. And \(p\): \(5x + 13y\) is prime. By Law of Detachment, the conclusion is \(5x+13y\) cannot be factored.

Step4: Solve problem 3

Given \(p
ightarrow q\): If you visit Paris, then you will see the Eiffel Tower. And \(
eg q\): You did not see the Eiffel Tower. This is modus tollens (\(
eg q
ightarrow
eg p\)), not Law of Detachment. So no valid conclusion using Law of Detachment.

Step5: Solve problem 4

Given \(p
ightarrow q\): If the measure of an angle is greater than \(90^{\circ}\), then it is an obtuse angle. And \(p\): \(m\angle PQR=119^{\circ}\) (so \(p\) is true since \(119^{\circ}>90^{\circ}\)). By Law of Detachment, the conclusion is \(\angle PQR\) 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

Given \(p
ightarrow q\): If Nicole is tardy to class again, she will get detention. And \(q
ightarrow r\): If Nicole gets detention, her mother will take away her phone. By Law of Syllogism, the conclusion is If Nicole is tardy to class again, her mother will take away her phone.

Step8: Solve problem 6

Given \(p
ightarrow q\): If it is raining, then I will bring my umbrella. And \(
eg q
ightarrow
eg r\): If I do not bring my umbrella, then I do not need my rain - boots. We cannot directly apply Law of Syllogism. So no valid conclusion using Law of Syllogism.

Step9: Solve problem 7

Given \(p
ightarrow q\): If a natural number ends in 0, then it is divisible by 10. And \(q
ightarrow r\): If a natural number is divisible by 10, then it is divisible by 5. By Law of Syllogism, the conclusion is 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 using Law of Detachment.
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 using Law of Syllogism.
7. If a natural number ends in 0, then it is divisible by 5.