What is demorgan's law?

DeMorgan's Laws are a pair of rules in <a href="https://www.wikiwhat.page/kavramlar/Boolean%20algebra">Boolean algebra</a> and set theory that relate the logical operators AND and OR through negation. They provide a way to simplify or transform logical expressions.

Here's a breakdown:

• First Law: The negation of a conjunction (AND) is the disjunction (OR) of the negations.

  • In propositional logic: ¬(A ∧ B) ≡ (¬A ∨ ¬B)
  • In set theory: (A ∩ B)^c = A^c ∪ B^c
  • This means "the complement of the intersection of A and B is equal to the union of the complement of A and the complement of B".

• Second Law: The negation of a disjunction (OR) is the conjunction (AND) of the negations.

  • In propositional logic: ¬(A ∨ B) ≡ (¬A ∧ ¬B)
  • In set theory: (A ∪ B)^c = A^c ∩ B^c
  • This means "the complement of the union of A and B is equal to the intersection of the complement of A and the complement of B".

Key Concepts:

• <a href="https://www.wikiwhat.page/kavramlar/Negation">Negation</a>: Reversing the truth value of a statement (e.g., "not A").
• <a href="https://www.wikiwhat.page/kavramlar/Conjunction">Conjunction</a>: A logical AND operation (both conditions must be true).
• <a href="https://www.wikiwhat.page/kavramlar/Disjunction">Disjunction</a>: A logical OR operation (at least one condition must be true).
• <a href="https://www.wikiwhat.page/kavramlar/Set%20Theory">Set Theory</a>: A branch of mathematics that studies sets, which are collections of objects.
• <a href="https://www.wikiwhat.page/kavramlar/Complement%20(Set%20Theory)">Complement (Set Theory)</a>: Elements not in a set A are in its complement A^c.
• <a href="https://www.wikiwhat.page/kavramlar/Intersection%20(Set%20Theory)">Intersection (Set Theory)</a>: The intersection of sets A and B (A ∩ B) contains elements common to both A and B.
• <a href="https://www.wikiwhat.page/kavramlar/Union%20(Set%20Theory)">Union (Set Theory)</a>: The union of sets A and B (A ∪ B) contains all elements that are in A or B (or both).

Applications:

DeMorgan's Laws are used in:

• Simplifying logical expressions in computer science.
• Proving theorems in mathematics.
• Designing digital circuits.
• Database queries.