Math Problem Statement

Assignment 1 _Ananya.pdf

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solve

Solution

The assignment file contains a set of problems on topics related to logic, predicate logic, and automated theorem proving. Let's solve one of the problems mentioned.

Problem 26: Logical Equivalence

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

Solution:

To prove that ¬p → (q → r) and q → (p ∨ r) are logically equivalent, we'll use truth tables.

Let's break down both expressions:

  1. Expression 1: ¬p → (q → r)

    • The implication ¬p → (q → r) means "If ¬p is true, then q → r must be true."
    • The truth table for q → r is:
      • True when q is False or r is True.
  2. Expression 2: q → (p ∨ r)

    • The implication q → (p ∨ r) means "If q is true, then p ∨ r must be true."
    • The truth table for p ∨ r is:
      • True when at least one of p or r is true.

Let's now construct the truth tables:

pqr¬pq → r¬p → (q → r)p ∨ rq → (p ∨ r)
TTTFTTTT
TTFFFTTT
TFTFTTTT
TFFFTTFT
FTTTTTTT
FTFTFFFF
FFTTTTTT
FFFTTTFT

Analysis:

The columns for ¬p → (q → r) and q → (p ∨ r) are identical, meaning these two expressions are logically equivalent.

Conclusion:

Therefore, ¬p → (q → r) and q → (p ∨ r) are indeed logically equivalent.

Would you like any further details or have any questions?

Here are some related problems for practice:

  1. Prove the logical equivalence between ¬(p ∨ q) and ¬p ∧ ¬q.
  2. Show that (p ∧ q) → r is logically equivalent to p → (q → r).
  3. Determine if ¬(p ∧ q) and ¬p ∨ ¬q are logically equivalent.
  4. Prove that p → q is equivalent to ¬p ∨ q.
  5. Show that ¬(p → q) is equivalent to p ∧ ¬q.

Tip: When proving logical equivalences, truth tables are a straightforward method to verify the results, but algebraic manipulations using logical identities can provide deeper insight.