[0/4 Points] DETAILS MY NOTES Construct a truth table to determine which pairs of statements are equivalent. (Select all that apply. i. The sky not being clear is sufficient for it to be raining. ii. The sky being clear is sufficient for it not to be raining. iii. The sky not being clear is necessary for it to be raining. iv. The sky being clear is necessary for it not to be raining.

To analyze the statements and determine which pairs are equivalent, we can first translate each statement into logical expressions. Let's define: - \( P \): The sky is clear. - \( Q \): It is raining. Now we can express each statement: 1. **i. The sky not being clear is sufficient for it to be raining.** - This can be expressed as: \( \neg P \rightarrow Q \) 2. **ii. The sky being clear is sufficient for it not to be raining.** - This can be expressed as: \( P \rightarrow \neg Q \) 3. **iii. The sky not being clear is necessary for it to be raining.** - This can be expressed as: \( Q \rightarrow \neg P \) (If it is raining, then the sky is not clear) 4. **iv. The sky being clear is necessary for it not to be raining.** - This can be expressed as: \( \neg Q \rightarrow P \) (If it is not raining, then the sky is clear) Now we can construct a truth table for these statements. We will evaluate the truth values of \( P \) and \( Q \) and then determine the truth values of each statement. | \( P \) | \( Q \) | \( \neg P \) | \( \neg Q \) | \( \neg P \rightarrow Q \) | \( P \rightarrow \neg Q \) | \( Q \rightarrow \neg P \) | \( \neg Q \rightarrow P \) | |---------|---------|---------------|---------------|-----------------------------|----------------------------|-----------------------------|-----------------------------| | T | T | F | F | T | F | T | F | | T | F | F | T | T | T | T | T | | F | T | T | F | T | F | F | F | | F | F | T | T | T | T | T | T | Now, let's summarize the truth values for each statement: - **i. \( \neg P \rightarrow Q \)**: T, T, T, T (always true) - **ii. \( P \rightarrow \neg Q \)**: F, T, F, T (not always true) - **iii. \( Q \rightarrow \neg P \)**: T, T, F, T (not always true) - **iv. \( \neg Q \rightarrow P \)**: F, T, F, T (not always true) Now we can compare the truth values of the statements: - **i** is always true (T). - **ii** is not always true (F, T, F, T). - **iii** is not always true (T, T, F, T). - **iv** is not always true (F, T, F, T). From the truth table, we can see that: - **i** is equivalent to **iv** (both are always true). - **ii** is not equivalent to any of the others. - **iii** is not equivalent to any of the others. Thus, the pairs of statements that are equivalent are: - **i and iv**.