De Morgan
De Morgan's laws are a pair of fundamental rules in Boolean algebra and mathematical logic that describe how to express the negation of a conjunction and the negation of a disjunction. These laws are named after Augustus De Morgan, a 19th-century British mathematician and logician.
In formal notation, De Morgan's laws are expressed as follows:
- ¬(A ∧ B) ≡ (¬A ∨ ¬B)
- ¬(A ∨ B) ≡ (¬A ∧ ¬B)
Where:
- ¬ represents negation (NOT)
- ∧ represents conjunction (AND)
- ∨ represents disjunction (OR)
- A and B are logical statements or sets.
- ≡ represents logical equivalence.
Interpretation:
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The first law states that the negation of a conjunction (A AND B) is logically equivalent to the disjunction of the negations of A and B (NOT A OR NOT B). In simpler terms, if it's not the case that both A and B are true, then at least one of them must be false.
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The second law states that the negation of a disjunction (A OR B) is logically equivalent to the conjunction of the negations of A and B (NOT A AND NOT B). This means that if it's not the case that either A or B is true, then both A and B must be false.
Applications:
De Morgan's laws are widely used in various fields including:
- Computer Science: Simplifying Boolean expressions in digital circuit design and programming. They are crucial for optimizing code and hardware implementations.
- Mathematics: Proof simplification and logical reasoning.
- Set Theory: Manipulating set operations involving complements, unions, and intersections.
- Logic: Rewriting and simplifying logical statements.
- Database Query Optimization: Rewriting SQL queries for improved performance.
Essentially, De Morgan's laws provide a way to transform expressions involving ANDs, ORs, and NOTs, enabling more efficient or understandable representations. They are a core concept in understanding and manipulating logical and Boolean expressions.