show that p$$ \lrArr\lnot q\text{ does not logically imply p}\rarr q $$

To show that $ p \iff \neg q $ does not logically imply $ p \rightarrow q $, we'll need to demonstrate that there are situations (i.e., truth assignments) where $ p \iff \neg q $ is true, but $ p \rightarrow q $ is false. We'll do this by examining the truth tables for both expressions and finding a case where the first is true and the second is false.

### Truth table construction

Let's define the expressions:

1. $ p \iff \neg q $ means that $ p $ is true if and only if $ q $ is false. This is the biconditional.
2. $ p \rightarrow q $ means that if $ p $ is true, then $ q $ must also be true. This is the implication.

We'll construct a truth table for both of these statements and compare their truth values.

| $ p $ | $ q $ | $ \neg q $ | $ p \iff \neg q $ | $ p \rightarrow q $ |
|--------|--------|-------------|---------------------|----------------------|
| T      | T      | F           | F                   | T                    |
| T      | F      | T           | T                   | F                    |
| F      | T      | F           | T                   | T                    |
| F      | F      | T           | F                   | T                    |

### Analyzing the table:

- In the second row, we have $ p = T $ and $ q = F $:
  - $ p \iff \neg q $ is **true** because $ p = T $ and $ \neg q = T $.
  - $ p \rightarrow q $ is **false** because $ p = T $ but $ q = F $.

This row shows that there exists a situation where $ p \iff \neg q $ is true, but $ p \rightarrow q $ is false. Therefore, we have proven that:

\[
p \iff \neg q \not\implies p \rightarrow q.
\]

### Conclusion:

The statement $ p \iff \neg q $ does not logically imply $ p \rightarrow q $, as demonstrated by the truth table where the first can be true while the second is false.

Would you like further details or clarification?

Here are five related questions to deepen your understanding:
1. What is the difference between biconditional ($ \iff $) and implication ($ \rightarrow $)?
2. Can $ p \iff \neg q $ ever imply $ p \rightarrow q $ under specific circumstances?
3. How does the truth table for $ p \rightarrow q $ change when $ p $ is always false?
4. What is the meaning of logical equivalence between two statements?
5. How can you negate an implication?

**Tip:** Always compare truth tables when analyzing logical implications, as they provide a straightforward way to check for validity.

show that p ⟺ ¬q does not logically imply p → q

Explore the logical distinction between p ⟺ ¬q and p → q by constructing truth tables. Understand how p ⟺ ¬q does not logically imply p → q with step-by-step reasoning. Suitable for high school students in Grades 11-12 studying propositional logic.

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