Question

What is duality law in set theory?

Answer 1

The duality law in set theory refers to a principle where certain operations in set theory have corresponding dual operations that produce equivalent results when the roles of union (∪) and intersection (∩) are swapped. It helps in simplifying or understanding the behavior of sets in a more symmetric way.

Here’s a simple explanation:

1. The dual of a set expression is formed by replacing:
   

  - Union (∪) with Intersection (∩),
  - Intersection (∩) with Union (∪),
  - The universal set (U) with the empty set (∅),
  - The empty set (∅) with the universal set (U).

Example of Duality in Set Theory:

1. Original Law:  
   

   \[(A \cup B)′ = A′ \cap B′\]  
   This is De Morgan's Law, where the complement of a union is the intersection of the complements.

1. Dual Law:  
   

   \[(A \cap B)′ = A′ \cup B′\]  
   This is the dual of the above law, where the complement of an intersection is the union of the complements.

Why is it useful?

Duality helps in understanding the symmetry between different operations in set theory, which can make certain proofs or problem-solving easier. It’s also a key idea in Boolean algebra, which has many applications in logic, circuit design, and computer science.