Math Problem Statement

Prove that p ∧ (q ∨ r) = (p ∨ q) ∨ (p ∧ r).

Solution

The problem provided in the image asks to prove the following logical equivalence:

p(qr)=(pq)(pr)p \land (q \lor r) = (p \lor q) \lor (p \land r)

Here is the proof using logical equivalences:


Left-hand side: p(qr)p \land (q \lor r)

  1. Distributive Law: Expand p(qr)p \land (q \lor r) using the distributive property: p(qr)=(pq)(pr)p \land (q \lor r) = (p \land q) \lor (p \land r)

Right-hand side: (pq)(pr)(p \lor q) \lor (p \land r)

  1. Associativity of \lor: Rearrange the right-hand side (associativity allows this): (pq)(pr)=p(q(pr))(p \lor q) \lor (p \land r) = p \lor (q \lor (p \land r))

  2. Absorption Law: Simplify q(pr)q \lor (p \land r): q(pr)=(qp)(qr)q \lor (p \land r) = (q \lor p) \land (q \lor r) So: (pq)(pr)=p((qp)(qr))(p \lor q) \lor (p \land r) = p \lor ((q \lor p) \land (q \lor r))

  3. Idempotence and Absorption Laws: Simplify further:

    • qp=pqq \lor p = p \lor q (commutativity of \lor).
    • Thus: p((qp)(qr))=(pq)(pr)p \lor ((q \lor p) \land (q \lor r)) = (p \lor q) \land (p \lor r)

Now both sides match, completing the proof:

p(qr)=(pq)(pr)p \land (q \lor r) = (p \lor q) \lor (p \land r)


Would you like me to explain any specific step or provide further examples? 😊


Related Questions:

  1. What are the basic logical equivalences and how are they applied in proofs?
  2. How does the distributive law help simplify logical expressions?
  3. Can you prove the commutative and associative properties of \lor and \land?
  4. What is the significance of the absorption law in logical simplifications?
  5. How can logical equivalences be applied in computer science (e.g., circuit design)?

Tip:

Always simplify logical expressions step by step using standard equivalences like distributive, associative, and commutative laws for clarity and accuracy.

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Logic
Logical Equivalences

Formulas

Distributive Law: p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)
Associative Law: (A ∨ B) ∨ C = A ∨ (B ∨ C)
Absorption Law: A ∨ (A ∧ B) = A

Theorems

Distributive Property
Associative Property of ∨ and ∧
Absorption Law

Suitable Grade Level

College-level Logic or High School Advanced Mathematics