Question

write the converse, inverse, and contrapositive of each conditional statement. find the truth - value of each.

1. probability if the probability of an event is 0.1, then the event is unlikely to occur.
2. meteorology if freezing rain is falling, then the air temperature is 32°f or less. (hint: the freezing point of water is 32°f)

find the truth - value of each statement.

1. e lies in plane r.
2. $overleftrightarrow{cd}$ lies in plane f.
3. c, e, and d are coplanar.
4. plane f contains $overline{ed}$.
5. b and e are collinear.
6. $overleftrightarrow{bc}$ contains f and r.

draw a venn diagram.

Explanation:

22.

Step1: Define converse

Switch hypothesis and conclusion.
Converse: If an event is unlikely to occur, then the probability of an event is 0.1. This is false as an event can be unlikely with other probabilities (e.g., 0.05).

Step2: Define inverse

Negate both hypothesis and conclusion.
Inverse: If the probability of an event is not 0.1, then the event is not unlikely to occur. This is false as an event with probability 0.05 is still unlikely and not 0.1.

Step3: Define contra - positive

Negate and switch hypothesis and conclusion.
Contra - positive: If an event is not unlikely to occur, then the probability of an event is not 0.1. This is true as if an event is likely (not unlikely), its probability is not 0.1.

Step1: Define converse

Switch hypothesis and conclusion.
Converse: If the air temperature is 32°F or less, then freezing rain is falling. This is false as temperature can be 32°F or less without freezing rain (e.g., just snow or clear conditions).

Step2: Define inverse

Negate both hypothesis and conclusion.
Inverse: If freezing rain is not falling, then the air temperature is greater than 32°F. This is false as there could be other cold - weather conditions without freezing rain and temperature can still be 32°F or less.

Step3: Define contra - positive

Negate and switch hypothesis and conclusion.
Contra - positive: If the air temperature is greater than 32°F, then freezing rain is not falling. This is true because water freezes at 32°F and above that temperature, rain won't freeze.

Step1: Analyze 24

If E is in the shaded region representing plane $\mathcal{R}$, then it is true.

Step2: Analyze 25

If the line - segment $\overrightarrow{CD}$ lies entirely within the shaded region representing plane $\mathcal{F}$, then it is true.

Step3: Analyze 26

If points C, E, and D all lie on the same plane (either $\mathcal{R}$ or $\mathcal{F}$ or another plane that contains them), then they are coplanar and it is true.

Step4: Analyze 27

If the line - segment $\overrightarrow{ED}$ lies entirely within plane $\mathcal{F}$, then it is true.

Step5: Analyze 28

If points B and E lie on the same straight line, then they are collinear and it is true.

Step6: Analyze 29

If the line $\overleftrightarrow{BC}$ contains both planes $\mathcal{F}$ and $\mathcal{R}$, then it is true. But without more information about the exact position of the line and planes, we assume based on the diagram's visual cues.

Answer:

Converse: False; Inverse: False; Contra - positive: True

23.