Math Problem Statement

use logical identities to solve

Solution

Let's solve the problem using logical identities instead of a truth table.

Problem 26: Logical Equivalence

Statement: Show that ¬p → (q → r) and q → (p ∨ r) are logically equivalent.

Solution Using Logical Identities:

We need to prove that: ¬p(qr)q(pr)\neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \vee r)

Let's break this down step by step.

  1. Express the implications using logical equivalences:

    • ABA \rightarrow B is equivalent to ¬AB\neg A \vee B.

    So, the first expression becomes: ¬p(qr)¬(¬p)(qr)\neg p \rightarrow (q \rightarrow r) \equiv \neg (\neg p) \vee (q \rightarrow r)

    Simplifying the double negation: p(qr)p \vee (q \rightarrow r)

    Now, express qrq \rightarrow r using the equivalence qr¬qrq \rightarrow r \equiv \neg q \vee r: p(¬qr)p \vee (\neg q \vee r)

    This can be rewritten using the associative and commutative properties of \vee as: (p¬q)r(p \vee \neg q) \vee r

  2. Simplify the expression: We now have: p¬qrp \vee \neg q \vee r

    On the other side, express the second statement q(pr)q \rightarrow (p \vee r) using logical identities: q(pr)¬q(pr)q \rightarrow (p \vee r) \equiv \neg q \vee (p \vee r)

    Using the commutative property of \vee: ¬qpr\neg q \vee p \vee r

  3. Compare the expressions: Both expressions simplify to: p¬qrp \vee \neg q \vee r

    Since both sides reduce to the same expression, the original statements are logically equivalent.

Conclusion:

Therefore, ¬p(qr)\neg p \rightarrow (q \rightarrow r) and q(pr)q \rightarrow (p \vee r) are logically equivalent by using logical identities.

Would you like any further details or have any questions?

Here are some related problems for practice:

  1. Prove that p¬pp \vee \neg p is a tautology using logical identities.
  2. Show that ¬(pq)¬p¬q\neg(p \vee q) \equiv \neg p \wedge \neg q.
  3. Prove the equivalence (pq)rp(qr)(p \rightarrow q) \rightarrow r \equiv p \rightarrow (q \rightarrow r).
  4. Demonstrate that p¬pp \wedge \neg p is a contradiction.
  5. Simplify (pq)(¬pq)(p \vee q) \wedge (\neg p \vee q) using logical identities.

Tip: Understanding the basic logical identities, such as De Morgan's laws and the distributive, associative, and commutative properties, is crucial for simplifying and proving logical equivalences.

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Math Problem Analysis

Mathematical Concepts

Logical Equivalence
Logical Identities

Formulas

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Theorems

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Suitable Grade Level

Advanced