Consider the following proof:EE → P..∴ (P ∧ E) ∨ VWhich of the following is a derivation needed for this proof?P → EP ∨ EVP

To determine which derivation is needed for the proof $EE \rightarrow P$ leading to $(P \land E) \lor V$, we need to analyze the provided options.

Analysis of Options

1. Option 1: $P \rightarrow E$

   • This statement suggests that if $P$ is true, then $E$ is true. However, it does not directly contribute to deriving $(P \land E) \lor V$ from $EE \rightarrow P$.

2. Option 2: $P \lor E$

   • This standard disjunction indicates that either $P$ or $E$ is true. While it provides some flexibility, it does not directly assist in reaching the conclusion regarding $(P \land E) \lor V$.

3. Option 3: $P \lor V$

   • Here, we are stating that either $P$ is true or $V$ is true. This may seem somewhat related to the conclusion since if $P$ is valid, it can contribute to the complete disjunction needed for the proof’s conclusion.

Conclusion

Given that none of the options perfectly match the expected derivations required to connect $EE \rightarrow P$ to $(P \land E) \lor V$, we can infer that the more probable connection might be through $P \lor V$, indicating the acceptance of either being the case could be essential for reaching the final conclusion.

Final Answer

The derivation needed for this proof is $P \lor V$.