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: ______ 2. given: if a polynomial is prime, then it cannot be factored. 5x + 13y is prime. conclusion: ____ 3. given: if you visit paris, then you will see the eiffel tower. you did not see the eiffel tower. conclusion: ____ 4. 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. 5. given: if nicole is tardy to class again, she will get detention. if nicole gets detention, her mother will take away her phone. conclusion: ____ 6. 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: ____ 7. 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

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

Step3: Solve problem 2

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

Step4: Solve problem 3

The statement is "If you visit Paris (\(p\)), then you will see the Eiffel Tower (\(q\))" and you did not see the Eiffel Tower (\(
eg q\)). The Law of Detachment cannot be applied as we have \(
eg q\) instead of \(p\). So, no valid conclusion.

Step5: Solve problem 4

The statement is "If the measure of an angle is greater than \(90^{\circ}\) (\(p\)), then it is an obtuse angle (\(q\))" and \(m\angle PQR=115^{\circ}\) (\(p\) is true). So, \(\angle PQR\) 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

We have "If Nicole is tardy to class again (\(p\)), she will get detention (\(q\))" and "If Nicole gets detention (\(q\)), her mother will take away her phone (\(r\))". So, if Nicole is tardy to class again, her mother will take away her phone.

Step8: Solve problem 6

We have "If it is raining (\(p\)), then I will bring my umbrella (\(q\))" and "If I do not bring my umbrella (\(
eg q\)), then I do not need my rain boots (\(r\))". We cannot directly apply the Law of Syllogism as the statements are not in the correct \(p
ightarrow q\) and \(q
ightarrow r\) form for a straightforward conclusion. So, no valid conclusion.

Step9: Solve problem 7

We have "If a natural number ends in 0 (\(p\)), then it is divisible by 10 (\(q\))" and "If a natural number is divisible by 10 (\(q\)), then it is divisible by 5 (\(r\))". So, 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.