To establish the equivalence $p \rightarrow ((p \rightarrow q) \land \sim (\sim q \lor \sim p)) \equiv \sim p \lor (q \land p)$, we will manipulate the left-hand side step by step using the laws of logic.

1. Express the Implication: The implication $p \rightarrow A$ is equivalent to $\sim p \lor A$. $p \rightarrow ((p \rightarrow q) \land \sim (\sim q \lor \sim p)) \equiv \sim p \lor ((p \rightarrow q) \land \sim (\sim q \lor \sim p))$

2. Expand the First Implication: Rewrite $p \rightarrow q$: $p \rightarrow q \equiv \sim p \lor q$ Substituting this into the expression gives: $\sim p \lor ((\sim p \lor q) \land \sim (\sim q \lor \sim p))$

3. Simplify the Negation: The expression $\sim (\sim q \lor \sim p)$ can be simplified using De Morgan's laws: $\sim (\sim q \lor \sim p) \equiv q \land p$ Now substitute this back in: $\sim p \lor ((\sim p \lor q) \land (q \land p))$

4. Distribute: Apply distribution of conjunction over disjunction: $(\sim p \lor q) \land (q \land p) \equiv \sim p \lor (q \land p)$ Hence, we have: $\sim p \lor (q \land p)$

5. Final Step: The expression is now simplified to: $\sim p \lor (q \land p)$ This matches the right-hand side of our original equivalence: $\sim p \lor (q \land p)$

Thus, we have established the equivalence: $p \rightarrow ((p \rightarrow q) \land \sim (\sim q \lor \sim p)) \equiv \sim p \lor (q \land p)$

Summary

We used logical equivalences and laws to transform the left side step-by-step until we matched the right side.

Do you have any questions or would you like further details on any part of this? Here are some related questions:

1. What are the laws of logic used in the proof?
2. Can you explain De Morgan's laws in detail?
3. How do we interpret logical implications in propositional logic?
4. What are some common examples of logical equivalences?
5. How can truth tables be used to verify this equivalence?

Tip: When working with logical equivalences, writing out each step clearly helps prevent mistakes and aids in understanding the transformations.