Questions: Write the converse, inverse, and contrapositive of the following proposition. Of these four propositions, state which pairs are equivalent. If the sun is shining, then it is wet outside. Identify the converse. Choose the correct answer below. A. If the sun is not shining, then it is not wet outside. B. If it is not wet outside, then the sun is not shining. C. If it is wet outside, then the sun is shining. D. If the sun is shining, then it is not wet outside. Identify the inverse. Choose the correct answer below. A. If the sun is shining, then it is not wet outside. B. If the sun is not shining, then it is not wet outside. C. If it is not wet outside, then the sun is not shining. D. If it is wet outside, then the sun is shining.

Write the converse, inverse, and contrapositive of the following proposition. Of these four propositions, state which pairs are equivalent. If the sun is shining, then it is wet outside.

Identify the converse. Choose the correct answer below.
A. If the sun is not shining, then it is not wet outside.
B. If it is not wet outside, then the sun is not shining.
C. If it is wet outside, then the sun is shining.
D. If the sun is shining, then it is not wet outside.

Identify the inverse. Choose the correct answer below.
A. If the sun is shining, then it is not wet outside.
B. If the sun is not shining, then it is not wet outside.
C. If it is not wet outside, then the sun is not shining.
D. If it is wet outside, then the sun is shining.

Transcript text: Write the converse, inverse, and contrapositive of the following proposition. Of these four propositions, state which pairs are equivalent. If the sun is shining, then it is wet outside. Identify the converse. Choose the correct answer below. A. If the sun is not shining, then it is not wet outside. B. If it is not wet outside, then the sun is not shining. C. If it is wet outside, then the sun is shining. D. If the sun is shining, then it is not wet outside. Identify the inverse. Choose the correct answer below. A. If the sun is shining, then it is not wet outside. B. If the sun is not shining, then it is not wet outside. C. If it is not wet outside, then the sun is not shining. D. If it is wet outside, then the sun is shining.

Solution

Solution Steps

To solve this problem, we need to understand the definitions of converse, inverse, and contrapositive for a given conditional statement. The original proposition is "If the sun is shining, then it is wet outside." The converse is formed by swapping the hypothesis and conclusion. The inverse is formed by negating both the hypothesis and conclusion. The contrapositive is formed by both swapping and negating the hypothesis and conclusion. We then identify which pairs of these propositions are logically equivalent.

Step 1: Identify the Original Proposition

The original proposition is given as: \[ P: \text{If the sun is shining, then it is wet outside.} \]

Step 2: Determine the Converse

The converse of the proposition \( P \) is formed by swapping the hypothesis and conclusion: \[ C: \text{If it is wet outside, then the sun is shining.} \]

Step 3: Determine the Inverse

The inverse of the proposition \( P \) is formed by negating both the hypothesis and conclusion: \[ I: \text{If the sun is not shining, then it is not wet outside.} \]

Step 4: Determine the Contrapositive

The contrapositive of the proposition \( P \) is formed by both swapping and negating the hypothesis and conclusion: \[ CP: \text{If it is not wet outside, then the sun is not shining.} \]

Step 5: Identify Equivalent Pairs

From the analysis, we find that:

The original proposition \( P \) is equivalent to its contrapositive \( CP \).
The converse \( C \) is equivalent to the inverse \( I \).

Final Answer

The answers to the multiple-choice questions are:

The converse is \( C: \text{If it is wet outside, then the sun is shining.} \)
The inverse is \( I: \text{If the sun is not shining, then it is not wet outside.} \)

Thus, the final answers are: \[ \boxed{C} \quad \text{(Converse)} \] \[ \boxed{I} \quad \text{(Inverse)} \]

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