https://file.aiquickdraw.com/mathbot/file/1725387858256edln9s4b.jpg

Let's break down the argument given in the image into symbolic logic and then test its validity using a truth table.

### Step 1: Translating Statements into Symbols

Let's assign symbols to each statement:

- Let \( p \): "The weather is sunny."
- Let \( q \): "The picnic will be enjoyable."

Now translate each statement:
1. "If the weather is sunny, then the picnic will be enjoyable."  
   This can be symbolized as \( p \rightarrow q \).

2. "If the picnic is enjoyable, then the weather was not sunny."  
   This can be symbolized as \( q \rightarrow \neg p \).

3. "The weather was not sunny and the picnic was not enjoyable."  
   This can be symbolized as \( \neg p \land \neg q \).

### Step 2: Argument Structure

We want to check the validity of the following argument:
- Premise 1: \( p \rightarrow q \)
- Premise 2: \( q \rightarrow \neg p \)
- Conclusion: \( \neg p \land \neg q \)

### Step 3: Constructing a Truth Table

To test the validity of this argument, we construct a truth table for all possible truth values of \( p \) and \( q \).

| \( p \) | \( q \) | \( p \rightarrow q \) | \( \neg p \) | \( q \rightarrow \neg p \) | \( \neg p \land \neg q \) |
|---------|---------|-----------------------|--------------|----------------------------|---------------------------|
| T       | T       | T                     | F            | F                          | F                         |
| T       | F       | F                     | F            | T                          | F                         |
| F       | T       | T                     | T            | T                          | F                         |
| F       | F       | T                     | T            | T                          | T                         |

### Step 4: Analyzing the Truth Table

- **Row 1:** Both premises \( p \rightarrow q \) and \( q \rightarrow \neg p \) are true, but the conclusion \( \neg p \land \neg q \) is false.
- **Row 2:** \( p \rightarrow q \) is false, so this row does not affect the argument's validity since we are only concerned when all premises are true.
- **Row 3:** Both premises are true, but the conclusion is false.
- **Row 4:** All statements, including the conclusion, are true.

### Conclusion from Truth Table

Since there is at least one case (specifically Row 1 and Row 3) where both premises are true, but the conclusion is false, the argument is **not valid**. 

The truth table shows that the argument does not hold in all cases, therefore the conclusion does not necessarily follow from the premises.

Would you like more details or have any questions about this? Here are some related questions:

1. How do we determine the validity of an argument using truth tables?
2. Can an argument be valid if there is any instance where the premises are true, but the conclusion is false?
3. How does negation affect logical statements in truth tables?
4. What are other methods to check for logical validity besides truth tables?
5. How can we apply this logic to more complex statements?

**Tip:** When using truth tables, ensure all possible truth values are considered for the variables involved to accurately determine the argument's validity.

Use symbols to write the logical form of each argument given below, and then use a truth table to test the argument for validity. 1. If the weather is sunny, then the picnic will be enjoyable. 2. If the picnic is enjoyable, then the weather was not sunny. 3. The weather was not sunny and the picnic was not enjoyable.

Learn how to translate arguments into symbolic logic and test their validity using truth tables. This problem involves concepts such as implications, negations, and conjunctions, and is suitable for students in Grades 10-12.

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