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The image contains a series of logical statements written in symbolic form. Let's analyze and break down the logical expressions step by step.

1. **\[(\sim q \vee r) \Rightarrow q\] \Leftrightarrow \[(p \land \neg r)\] \Leftrightarrow [(\sim q \vee p) \land (\sim q \vee r)]**
   - This expression suggests that the statement \[(\sim q \vee r) \Rightarrow q\] is logically equivalent to \[(p \land \neg r)\], and both are equivalent to \[(\sim q \vee p) \land (\sim q \vee r)\].

2. **\[(p \Rightarrow q) \land (\sim p \Rightarrow q)\] \Rightarrow q**
   - Here, the conjunction of two implications, \((p \Rightarrow q)\) and \((\sim p \Rightarrow q)\), leads to the conclusion \(q\).

3. **\[(p \Rightarrow q) \land (p \Rightarrow \neg q)\] \Leftrightarrow [p \Rightarrow (q \land \neg r)]**
   - This states that if \(p\) implies both \(q\) and \(\neg q\), then \(p\) must imply \(q \land \neg r\).

4. **\[(p \Leftrightarrow q) \land (p \land Np)\] \Leftrightarrow N(q \Rightarrow p)**
   - This expression involves a biconditional \((p \Leftrightarrow q)\) and a contradiction \((p \land Np)\), leading to the negation of the implication \(q \Rightarrow p\).

5. **\[(p \land \sim q) \lor (p \land r)\] is the negation of \(\sim (p \Rightarrow q)\)**
   - The disjunction \((p \land \sim q) \lor (p \land r)\) represents the negation of the implication \(\sim (p \Rightarrow q)\).

6. **\[(p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow \sim p)\]**
   - This is the statement of the contrapositive, where the implication \(p \Rightarrow q\) is equivalent to its contrapositive \(\sim q \Rightarrow \sim p\).

7. **\[(\sim (p \land q)) \Rightarrow r\] \Leftrightarrow [\sim [(p \Rightarrow \sim q) \land \sim r]]**
   - The negation of the conjunction \((p \land q)\) leading to \(r\) is equivalent to the negation of the conjunction \((p \Rightarrow \sim q) \land \sim r\).

These statements involve logical equivalences and implications in propositional logic. They can be verified using truth tables or logical equivalence rules.

Would you like details on how to construct truth tables for these expressions, or do you have any other questions?

Here are five related questions:
1. How do you construct a truth table for a complex logical expression?
2. What are the common logical equivalence rules in propositional logic?
3. How can you prove logical equivalences using algebraic manipulation of logical expressions?
4. What is the significance of the contrapositive in logical implications?
5. How does the negation of a conjunction or disjunction affect the logical equivalence of statements?

**Tip:** When dealing with logical statements, using truth tables is a systematic way to verify the validity and equivalence of different expressions.

Explore detailed explanations and examples of logical equivalences in propositional logic. Learn how to construct truth tables for complex logical expressions and verify their validity. Ideal for students and professionals interested in advanced logical reasoning and problem-solving.

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