Math Problem Statement
Show that ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by developing a series of logical equivalences
Solution
To show that and are logically equivalent, we will develop a series of logical equivalences.
Step 1: Apply De Morgan’s Law
Using De Morgan’s Law: , we rewrite:
Step 2: Apply De Morgan’s Law Again
Using De Morgan’s Law: , we rewrite:
Since , this simplifies to:
Step 3: Apply Distribution
Using the Distributive Law: , we expand:
Since is always false (), we get:
Since , we obtain:
Conclusion:
Thus, we have shown that:
This proves that the two statements are logically equivalent.
Would you like a more detailed breakdown or another example? Here are some related questions to explore:
- Can you prove that is equivalent to ?
- How does the principle of double negation apply in logical proofs?
- Can you show the equivalence of and ?
- What is the difference between a tautology and a contradiction in logic?
- How do truth tables help verify logical equivalences?
Tip: When working with logical equivalences, always apply De Morgan’s Laws and distribution carefully to simplify expressions systematically.
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Logical Equivalence
Boolean Algebra
Formulas
De Morgan's Laws: ¬(A ∨ B) ≡ ¬A ∧ ¬B, ¬(A ∧ B) ≡ ¬A ∨ ¬B
Distributive Law: A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C)
Theorems
De Morgan's Laws
Distributive Law
Double Negation
Suitable Grade Level
Undergraduate (Discrete Mathematics) or Advanced High School (Grade 11-12)
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