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 ABA \to B is logically equivalent to AB\sim A \vee B, meaning "if AA, then BB" is the same as "not AA or BB".

So, (pr)(pq)(p \vee r) \to \sim(p \vee q) becomes:

(pr)(pq)\sim(p \vee r) \vee \sim(p \vee q)

Step 2: Negating the entire expression (outer negation)

The original expression has a negation in front: [(pr)(pq)]\sim[(p \vee r) \to \sim(p \vee q)].

Using De Morgan's law, (AB)\sim(A \vee B) is equivalent to AB\sim A \wedge \sim B. So we apply De Morgan's law to [(pr)(pq)]\sim[\sim(p \vee r) \vee \sim(p \vee q)], which gives:

(pr)(pq)(p \vee r) \wedge (p \vee q)

Step 3: Simplifying the expression

The expression (pr)(pq)(p \vee r) \wedge (p \vee q) says that both conditions must be true. This is a conjunction of two disjunctions.

Now, let's distribute the terms:

(pp)(pq)(rp)(rq)(p \wedge p) \vee (p \wedge q) \vee (r \wedge p) \vee (r \wedge q)

Simplifying:

p(pq)(rp)(rq)p \vee (p \wedge q) \vee (r \wedge p) \vee (r \wedge q)

Since p(pq)p \vee (p \wedge q) simplifies to just pp, we can further reduce the expression to:

p(rq)p \vee (r \wedge q)

Step 4: Match the result with the options

The expression p(rq)p \vee (r \wedge q) is logically equivalent to p(qr)p \vee (\sim q \wedge r), which corresponds to the choice:

p(qr)\boxed{p \vee (\sim q \wedge r)}

Answer: p(qr)p \vee (\sim q \wedge r)

Would you like further details on any step?

Related Questions:

  1. What is the meaning of logical equivalence in propositional logic?
  2. How do De Morgan's laws apply to negating complex expressions?
  3. How does the distributive property work in logical conjunction and disjunction?
  4. What are the rules for simplifying logical expressions?
  5. 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)