Math Problem Statement
Which of the following statements is logically equivalent to ∼[(𝑝 ∨ 𝑟) → ∼(𝑝 ∨ 𝑞)]?
Group of answer choices
𝑝 ∨ (𝑞 ∧ 𝑟)
𝑝 ∧ (𝑞 ∨ 𝑟)
(𝑝 ∨ 𝑞) ∧ 𝑟
𝑝 ∨ (∼𝑞 ∧ 𝑟)
𝑝 ∧ (𝑞 ∨ ∼𝑟)
Solution
To find the logically equivalent statement to ∼[(𝑝 ∨ 𝑟) → ∼(𝑝 ∨ 𝑞)], let's break down the expression step by step.
Step 1: Recall the logical implication (→)
An implication is logically equivalent to , meaning "if , then " is the same as "not or ".
So, becomes:
Step 2: Negating the entire expression (outer negation)
The original expression has a negation in front: .
Using De Morgan's law, is equivalent to . So we apply De Morgan's law to , which gives:
Step 3: Simplifying the expression
The expression says that both conditions must be true. This is a conjunction of two disjunctions.
Now, let's distribute the terms:
Simplifying:
Since simplifies to just , we can further reduce the expression to:
Step 4: Match the result with the options
The expression is logically equivalent to , which corresponds to the choice:
Answer:
Would you like further details on any step?
Related Questions:
- What is the meaning of logical equivalence in propositional logic?
- How do De Morgan's laws apply to negating complex expressions?
- How does the distributive property work in logical conjunction and disjunction?
- What are the rules for simplifying logical expressions?
- How can truth tables help verify logical equivalence?
Tip:
Always simplify expressions by breaking them down step by step, especially when dealing with negations and implications.
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Logical Equivalence
De Morgan's Law
Implication in Logic
Formulas
Implication: A → B ≡ ¬A ∨ B
De Morgan's Law: ¬(A ∨ B) ≡ ¬A ∧ ¬B
Theorems
De Morgan's Law
Distributive Property of Logical Operations
Suitable Grade Level
Undergraduate (or Advanced High School Level)