Question

determine if each conditional is true. if false, give a counterexample.

1. if three points form the vertices of a triangle, then they lie in the same plane.
2. if (x > y), then (|x|>|y|).
3. if the season is spring, then the month is march.
4. travel write the converse, inverse, and contrapositive of the following conditional statement. find the truth - value of each.

if brielle drives at exactly 30 mi/h, then she travels 10 mi in 20 min.

Explanation:

Step1: Analyze statement 9

By the flat - plane property of triangles in geometry, any three non - collinear points (vertices of a triangle) determine a unique plane. So this statement is true.

Step2: Analyze statement 10

Let \(x = 1\) and \(y=-2\). Then \(x>y\) since \(1 > - 2\), but \(|x| = 1\) and \(|y|=2\), and \(|x|<|y|\). So this statement is false.

Step3: Analyze statement 11

Spring season in the Northern Hemisphere includes March, April, and May. Just because it is spring doesn't mean it is March only. So this statement is false.

Step4: Analyze statement 12 - Converse

The original statement is \(p
ightarrow q\) where \(p\): Brielle drives at exactly 30 mi/h and \(q\): she travels 10 mi in 20 min. The converse is \(q
ightarrow p\). If she travels 10 mi in 20 min, her speed \(v=\frac{d}{t}=\frac{10}{\frac{20}{60}} = 30\) mi/h. So the converse is true.

Step5: Analyze statement 12 - Inverse

The inverse of \(p
ightarrow q\) is \(
eg p
ightarrow
eg q\). If Brielle does not drive at exactly 30 mi/h, she may or may not travel 10 mi in 20 min. For example, if she drives at 60 mi/h, she travels 20 mi in 20 min. But if she drives at 15 mi/h, she travels 5 mi in 20 min. So the inverse is false.

Step6: Analyze statement 12 - Contrapositive

The contrapositive of \(p
ightarrow q\) is \(
eg q
ightarrow
eg p\). If she does not travel 10 mi in 20 min, her speed is not 30 mi/h. Since \(v = \frac{d}{t}\), if \(d
eq10\) mi or \(t
eq20\) min (in the relevant ratio), then \(v
eq30\) mi/h. So the contrapositive is true.

Answer:

1. True
2. False, counter - example: \(x = 1,y=-2\)
3. False
4. Converse: If Brielle travels 10 mi in 20 min, then she drives at exactly 30 mi/h. True. Inverse: If Brielle does not drive at exactly 30 mi/h, then she does not travel 10 mi in 20 min. False. Contrapositive: If Brielle does not travel 10 mi in 20 min, then she does not drive at exactly 30 mi/h. True.