Questions: Question Use a truth table to determine which of the following statements, if any, are equivalent to the statement "It is raining today and so my garden is getting watered today." - My garden's not getting watered today and it's not raining today. - My garden is getting watered today because it is raining today. - It's raining today, yet my garden isn't getting watered today. - Even though it's not raining today, my garden is getting watered today. Select the correct answer below: My garden's not getting watered today and it's not raining today My garden is getting watered today because it is raining today It's raining today, yet my garden isn't getting watered today Even though it's not raining today, my garden is getting watered today FEEDBACK MORE INSTRUCTION SUBMIT Content attribution

Solution Steps

To determine which statements are equivalent to the given statement using a truth table, we need to analyze the logical structure of each statement. The original statement is a conjunction ("and" statement), so we will compare it to each option to see if they have the same truth values under all possible scenarios.

The original statement is "It is raining today and so my garden is getting watered today." This can be expressed as a logical conjunction: \( R \land W \), where \( R \) represents "It is raining today" and \( W \) represents "My garden is getting watered today."

• Option 1: "My garden's not getting watered today and it's not raining today." This is expressed as \( \neg W \land \neg R \).
• Option 2: "My garden is getting watered today because it is raining today." This is expressed as \( R \land W \).
• Option 3: "It's raining today, yet my garden isn't getting watered today." This is expressed as \( R \land \neg W \).
• Option 4: "Even though it's not raining today, my garden is getting watered today." This is expressed as \( \neg R \land W \).

We evaluate the truth values for each statement under all possible scenarios of \( R \) and \( W \):

• \( (R, W) = (T, T) \)
• \( (R, W) = (T, F) \)
• \( (R, W) = (F, T) \)
• \( (R, W) = (F, F) \)

The truth values for each statement are:

• Original: \([T, F, F, F]\)
• Option 1: \([F, F, F, T]\)
• Option 2: \([T, F, F, F]\)
• Option 3: \([F, T, F, F]\)
• Option 4: \([F, F, T, F]\)

A statement is equivalent to the original if it has the same truth values for all scenarios. Comparing the truth values, we find:

• Option 2 has the same truth values as the original statement.

Final Answer