This is a propositional logic statement:

P → (Q → P ∧ Q)

Let’s solve it step-by-step:

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Step 1: Understand the logical formula

• P → (Q → P ∧ Q) means:
  "If P is true, then if Q is true, P ∧ Q must also be true."

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Step 2: Break it down using logical equivalences

1. Implication rewrite: The implication P → X is equivalent to ¬P ∨ X.
   Rewrite the formula:
   $¬P ∨ (¬Q ∨ (P ∧ Q))$

2. Distribute the parentheses:
   Simplify ¬Q ∨ (P ∧ Q) using the distributive property:
   $¬Q ∨ P ∨ Q$

   (This step is true because ¬Q ∨ (P ∧ Q) means either ¬Q is true, or P and Q are both true. Simplifying this leads to ¬Q ∨ P ∨ Q.)

3. Combine terms:
   Notice that ¬P ∨ (¬Q ∨ P ∨ Q) can be combined further:
   $¬P ∨ P ∨ ¬Q ∨ Q$

   • ¬P ∨ P simplifies to True (a tautology).
   • ¬Q ∨ Q also simplifies to True (a tautology).

   So the entire expression simplifies to True.

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Final Answer:

The statement P → (Q → P ∧ Q) is always True (a tautology).

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Would you like me to explain any specific part of the solution further?

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Related Questions:

1. What are the rules for simplifying logical expressions?
2. How does the implication P → Q translate into logical equivalence?
3. Why does ¬P ∨ P simplify to a tautology (True)?
4. What is the significance of a tautology in propositional logic?
5. How can truth tables be used to verify tautologies?

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Tip:

When solving logical expressions, always start by rewriting implications as disjunctions (¬P ∨ Q) to make simplifications easier!