The logical expression ¬q ⟹ ¬p can be analyzed using the rules of implication in propositional logic. Here's the detailed explanation:

### Expression Breakdown

1. **¬q**: This represents the negation of statement \( q \).
2. **¬p**: This represents the negation of statement \( p \).
3. **⟹**: This is the implication symbol, meaning "if ... then ...".

So, the expression ¬q ⟹ ¬p reads as:

**"If not \( q \), then not \( p \)."**

### Logical Equivalence

The expression ¬q ⟹ ¬p is logically equivalent to its contrapositive:

**\( p \) ⟹ \( q \)**

- This equivalence arises because an implication \( A \rightarrow B \) is always equivalent to its contrapositive \( \neg B \rightarrow \neg A \). 

### Truth Table Analysis

To further understand this, let's build a truth table for both expressions:

| \( p \) | \( q \) | ¬\( q \) | ¬\( p \) | ¬\( q \) ⟹ ¬\( p \) | \( p \) ⟹ \( q \) |
|---------|---------|-----------|-----------|-----------------------|--------------------|
| T       | T       | F         | F         | T                     | T                  |
| T       | F       | T         | F         | F                     | F                  |
| F       | T       | F         | T         | T                     | T                  |
| F       | F       | T         | T         | T                     | T                  |

From the table, you can observe that the truth values of ¬q ⟹ ¬p are the same as \( p \) ⟹ \( q \), confirming the equivalence.

### Conclusion

The statement ¬q ⟹ ¬p implies that whenever \( q \) is false, \( p \) must also be false. Equivalently, if \( p \) is true, \( q \) must be true.

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

1. How can you determine the contrapositive of a given implication?
2. What is the significance of truth tables in logic?
3. How does an implication differ from a biconditional statement?
4. Can you explain the converse and inverse of an implication?
5. How does the logical equivalence between implications simplify complex logical expressions?

**Tip:** Always check the contrapositive of an implication for a clearer understanding of logical equivalences!

The query submitted by the user: ¬q ⟹ ¬p

Explore the logical expression ¬q ⟹ ¬p and its contrapositive p ⟹ q in propositional logic. Learn about the rules of implication, contrapositive theorem, and truth table analysis. Suitable for undergraduate students studying logic.

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