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Show that ¬(𝑝 ∨ (¬𝑝 ∧ 𝑞)) and ¬𝑝 ∧ ¬𝑞 are logically equivalent by:a) Using a truth table (3 POINTS)b) Using logical equivalences

Question

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Solution

Sure, let's start with the truth table:

p q ¬p ¬q ¬p ∧ q p ∨ (¬p ∧ q) ¬(p ∨ (¬p ∧ q)) ¬p ∧ ¬q
T T F F F T F F
T F F T F T F F
F T T F T T F F
F F T T F F T T

As you can see from the truth table, ¬(𝑝 ∨ (¬𝑝 ∧ 𝑞)) and ¬𝑝 ∧ ¬𝑞 have the same truth values in all possible scenarios, so they are logically equivalent.

Now, let's prove this using logical equivalences:

  1. ¬(𝑝 ∨ (¬𝑝 ∧ 𝑞)) = ¬𝑝 ∧ ¬(¬𝑝 ∧ 𝑞) (De Morgan's Law)
  2. ¬𝑝 ∧ ¬(¬𝑝 ∧ 𝑞) = ¬𝑝 ∧ (𝑝 ∨ ¬𝑞) (De Morgan's Law)
  3. ¬𝑝 ∧ (𝑝 ∨ ¬𝑞) = (¬𝑝 ∧ 𝑝) ∨ (¬𝑝 ∧ ¬𝑞) (Distributive Law)
  4. (¬𝑝 ∧ 𝑝) ∨ (¬𝑝 ∧ ¬𝑞) = F ∨ (¬𝑝 ∧ ¬𝑞) (Law of Contradiction)
  5. F ∨ (¬𝑝 ∧ ¬𝑞) = ¬𝑝 ∧ ¬𝑞 (Identity Law)

So, ¬(𝑝 ∨ (¬𝑝 ∧ 𝑞)) and ¬𝑝 ∧ ¬𝑞 are logically equivalent.

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