Question

example 3
use the law of syllogism to draw a valid conclusion from each set of given statements, if possible. if no valid conclusion can be drawn, write no valid conclusion and explain your reasoning.

1. if you interview for a job, then you wear a suit.

if you interview for a job, then you will update your resume.

1. if tina has a grade point average of 3.0 or greater, she will be on the honor roll.

if tina is on the honor roll, then she will have her name in the school paper.

1. if two lines are perpendicular, then they intersect to form right angles.

lines s and r form right angles.

1. if the measure of an angle is between 90° and 180°, then it is obtuse.

if an angle is obtuse, then it is not acute.

1. if two lines in a plane are not parallel, then they intersect.

if two lines intersect, then they intersect in a point.

1. if a number ends in 0, then it is divisible by 2.

if a number ends in 4, then it is divisible by 2.

Explanation:

Step1: Recall Law of Syllogism

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

Step2: Analyze problem 12

Let \(p\) be "you interview for a job", \(q\) be "you wear a suit", and \(r\) be "you update your resume". The first statement is \(p
ightarrow q\) and the second is \(p
ightarrow r\). Since the middle - term (\(q\)) is not used in the correct transitive way for the Law of Syllogism, no valid conclusion.

Step3: Analyze problem 13

Let \(p\) be "Tina has a grade - point average of 3.0 or greater", \(q\) be "she will be on the honor roll", and \(r\) be "she will have her name in the school paper". We have \(p
ightarrow q\) and \(q
ightarrow r\). By the Law of Syllogism, if Tina has a grade - point average of 3.0 or greater, then she will have her name in the school paper.

Step4: Analyze problem 14

The first statement is \(p
ightarrow q\) where \(p\) is "two lines are perpendicular" and \(q\) is "they intersect to form right angles". The second statement just affirms that lines \(s\) and \(r\) satisfy \(q\). This is not in the form for the Law of Syllogism, no valid conclusion.

Step5: Analyze problem 15

Let \(p\) be "the measure of an angle is between 90° and 180°", \(q\) be "it is obtuse", and \(r\) be "it is not acute". We have \(p
ightarrow q\) and \(q
ightarrow r\). By the Law of Syllogism, if the measure of an angle is between 90° and 180°, then it is not acute.

Step6: Analyze problem 16

Let \(p\) be "two lines in a plane are not parallel", \(q\) be "they intersect", and \(r\) be "they intersect in a point". We have \(p
ightarrow q\) and \(q
ightarrow r\). By the Law of Syllogism, if two lines in a plane are not parallel, then they intersect in a point.

Step7: Analyze problem 17

Let \(p\) be "a number ends in 0", \(q\) be "it is divisible by 2", \(r\) be "a number ends in 4", and \(s\) be "it is divisible by 2". The statements are \(p
ightarrow q\) and \(r
ightarrow s\). Since there is no transitive relationship between the hypotheses and conclusions in the form required for the Law of Syllogism, no valid conclusion.

Answer:

1. No valid conclusion. The statements are not in the correct form for the Law of Syllogism.
2. If Tina has a grade - point average of 3.0 or greater, then she will have her name in the school paper.
3. No valid conclusion. The statements are not in the correct form for the Law of Syllogism.
4. If the measure of an angle is between 90° and 180°, then it is not acute.
5. If two lines in a plane are not parallel, then they intersect in a point.
6. No valid conclusion. The statements are not in the correct form for the Law of Syllogism.