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De Morgan's Law: A Comprehensive Proof of ¬(A ∪ B) = ¬A ∩ ¬B and ¬(A ∩ B) = ¬A ∪ ¬B

De Morgan's Law is a fundamental concept in Boolean algebra and logic, providing a crucial bridge between union and intersection operations. This article will provide a detailed, step-by-step proof of both parts of De Morgan's Law using set theory and truth tables, ensuring a clear understanding of its validity and application. Understanding De Morgan's Laws is essential for simplifying logical expressions and optimizing algorithms in computer science and digital electronics.

What are De Morgan's Laws?

De Morgan's Laws state that the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. Mathematically, this can be expressed as:

• ¬(A ∪ B) = ¬A ∩ ¬B (The complement of the union of A and B is equal to the intersection of the complement of A and the complement of B)
• ¬(A ∩ B) = ¬A ∪ ¬B (The complement of the intersection of A and B is equal to the union of the complement of A and the complement of B)

Where:

• A and B represent sets.
• ∪ represents the union (combination) of sets.
• ∩ represents the intersection (overlap) of sets.
• ¬ represents the complement (everything outside of the set).

Proof using Set Theory

We will prove ¬(A ∪ B) = ¬A ∩ ¬B. The proof for ¬(A ∩ B) = ¬A ∪ ¬B follows a similar structure.

1. Proving ¬(A ∪ B) ⊆ ¬A ∩ ¬B:

Let x be an arbitrary element in ¬(A ∪ B). By definition, this means x is not in (A ∪ B). Since x is not in (A ∪ B), it's neither in A nor in B. Therefore, x is in ¬A and x is in ¬B. Consequently, x is in the intersection of ¬A and ¬B (¬A ∩ ¬B). This proves that ¬(A ∪ B) ⊆ ¬A ∩ ¬B.

2. Proving ¬A ∩ ¬B ⊆ ¬(A ∪ B):

Now, let's assume x is an arbitrary element in ¬A ∩ ¬B. This implies that x is in ¬A and x is in ¬B. This means x is not in A and x is not in B. Therefore, x cannot be in the union of A and B (A ∪ B). Hence, x is in ¬(A ∪ B). This proves that ¬A ∩ ¬B ⊆ ¬(A ∪ B).

3. Conclusion:

Since we have shown that ¬(A ∪ B) ⊆ ¬A ∩ ¬B and ¬A ∩ ¬B ⊆ ¬(A ∪ B), we can conclude that ¬(A ∪ B) = ¬A ∩ ¬B. This completes the set theory proof for the first part of De Morgan's Law. The proof for ¬(A ∩ B) = ¬A ∪ ¬B follows the same logic, simply swapping the roles of union and intersection.

Proof using Truth Tables

Truth tables offer a more direct, albeit less elegant, approach to proving De Morgan's Laws. We'll demonstrate the proof for ¬(A ∪ B) = ¬A ∩ ¬B.

As you can see, the columns for ¬(A ∪ B) and ¬A ∩ ¬B are identical, demonstrating the equivalence of the two expressions. A similar truth table can be constructed to prove ¬(A ∩ B) = ¬A ∪ ¬B.

Applications of De Morgan's Laws

De Morgan's Laws are invaluable tools in various fields:

• Logic Simplification: They are essential for simplifying complex Boolean expressions, leading to more efficient digital circuits and algorithms.
• Set Theory: They provide a way to manipulate set operations and solve problems involving complements, unions, and intersections.
• Probability: They can be used to simplify probability calculations involving complementary events.

By understanding and applying De Morgan's Laws, you can significantly enhance your ability to work with logical expressions and set theory, leading to more efficient problem-solving across numerous domains.