Question

use the law of detachment or the law of syllogism to reach a valid conclusion. if a valid conclusion cannot be reached, write no conclusion.

1. given: if alex runs a mile in under 6 minutes, then he will have a new personal record.

given: alex ran a mile in 5 minutes and 45 seconds.
conclusion:________________________

1. given: if you go to the water park, then you will need to bring sunscreen.

given: if you do not bring sunscreen, then you will burn.
conclusion:________________________

1. given: if the measures of two angles in a triangle are 30° and 60°, then the third angle is 90°.

given: if a triangle has a 90° angle, then it is a right triangle.
conclusion:________________________

Explanation:

10. Step1: Check if the second statement satisfies the hypothesis of the first

The first statement is “If Alex runs a mile in under 6 minutes, then he will have a new personal record”. The second statement is “Alex ran a mile in 5 minutes and 45 seconds”. Since \(5\) minutes and \(45\) seconds (\(5 + \frac{45}{60}=5.75\) minutes) is under \(6\) minutes.

10. Step2: Apply the Law of Detachment

By the Law of Detachment (if \(p
ightarrow q\) is true and \(p\) is true, then \(q\) is true), where \(p\): “Alex runs a mile in under 6 minutes” and \(q\): “he will have a new personal record”

11. Step1: Rewrite the statements in symbolic form

Let \(p\): “you go to the water park”, \(q\): “you need to bring sunscreen”, \(r\): “you will burn”. The first statement is \(p
ightarrow q\), the second is \(
eg q
ightarrow r\). There is no direct way to use Law of Detachment or Law of Syllogism as we don't know if \(p\) is True.

12. Step1: Use the Law of Syllogism

Let \(p\): “the measures of two angles in a triangle are \(30^{\circ}\) and \(60^{\circ}\)”, \(q\): “the third angle is \(90^{\circ}\)”, \(r\): “it is a right - triangle”. The first statement is \(p
ightarrow q\), the second is \(q
ightarrow r\). By the Law of Syllogism (\(p
ightarrow q\) and \(q
ightarrow r\) implies \(p
ightarrow r\))

Answer:

1. Alex will have a new personal record.
2. No conclusion.
3. If the measures of two angles in a triangle are \(30^{\circ}\) and \(60^{\circ}\), then it is a right triangle.