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:
-
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.
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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:
p | q | r | ¬p | q → r | ¬p → (q → r) | p ∨ r | q → (p ∨ r) |
---|---|---|---|---|---|---|---|
T | T | T | F | T | T | T | T |
T | T | F | F | F | T | T | T |
T | F | T | F | T | T | T | T |
T | F | F | F | T | T | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | F | F | F |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | F | T |
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:
- Prove the logical equivalence between ¬(p ∨ q) and ¬p ∧ ¬q.
- Show that (p ∧ q) → r is logically equivalent to p → (q → r).
- Determine if ¬(p ∧ q) and ¬p ∨ ¬q are logically equivalent.
- Prove that p → q is equivalent to ¬p ∨ q.
- 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.
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Math Problem Analysis
Mathematical Concepts
Logic
Predicate Logic
Formulas
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Theorems
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Suitable Grade Level
Advanced