https://file.aiquickdraw.com/mathbot/file/1728058596061q53h3dl9.jpg

In the image, we have a logical expression \(\sim p \rightarrow \sim q\), which means "if not \(p\), then not \(q\)". You are likely trying to fill out a truth table for this logical expression based on the truth values of \(p\) and \(q\).

Here’s how we interpret the truth table step by step:

1. **Negation (~)**: The tilde (\(\sim\)) symbol represents negation, meaning \(\sim p\) is "not \(p\)" and \(\sim q\) is "not \(q\)".
2. **Implication (\(\rightarrow\))**: The arrow symbol (\(\rightarrow\)) is an implication, meaning "if the left side is true, then the right side must be true".

### Truth table construction:
Let's assume the possible truth values for \(p\) and \(q\) (True (T) or False (F)):

| \(p\) | \(q\) | \(\sim p\) | \(\sim q\) | \(\sim p \rightarrow \sim q\) |
|:----:|:----:|:---------:|:---------:|:---------------------------:|
|  T   |  T   |     F     |     F     |             T               |
|  T   |  F   |     F     |     T     |             T               |
|  F   |  T   |     T     |     F     |             F               |
|  F   |  F   |     T     |     T     |             T               |

### Explanation of the last column:
- When \(\sim p\) is false, the implication is always true regardless of \(\sim q\) (rows 1 and 2).
- When \(\sim p\) is true, the truth of the implication depends on \(\sim q\): the implication is true if \(\sim q\) is also true, and false otherwise.

Now, you can fill in the truth table boxes using this information.

Would you like more details on any step, or have further questions?

Here are some related questions:
1. What does the implication mean in logic?
2. How does negation (\(\sim\)) affect truth values?
3. Can we always assume that a false antecedent makes an implication true?
4. What is the difference between conjunction and implication in logic?
5. How do we handle compound logical statements in truth tables?

**Tip**: In logic, when the left-hand side of an implication is false, the entire implication is considered true, no matter the right-hand side.

Fill out the truth table for the logical expression 'not p implies not q'.

Learn how to construct a truth table for the logical expression 'not p implies not q'. This problem covers key concepts in propositional logic, including negation and logical implication. Suitable for college-level introductory logic students.

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