Algebra of sets | Laws in set theory

Basic Laws of Set Theory

Set theory provides a foundation for understanding how elements group and interact in mathematics. Here are the essential laws of set operations, explained in a simple and clear manner.

1. Idempotent Laws

• A ∪ A = A: Combining a set with itself using union results in the same set.
• A ∩ A = A: The intersection of a set with itself also gives the same set.

2. Commutative Laws

• A ∪ B = B ∪ A: Changing the order of sets in a union does not affect the outcome.
• A ∩ B = B ∩ A: The intersection does not change even if the order of sets is switched.

3. Associative Laws

• (A ∪ B) ∪ C = A ∪ (B ∪ C): Grouping of sets in a union does not affect the result.
• (A ∩ B) ∩ C = A ∩ (B ∩ C): Grouping does not affect the outcome of intersections either.

4. Distributive Laws

• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C): The intersection distributes over union.
• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C): Union distributes over intersection.

5. Identity Laws

• A ∪ ∅ = A: Union with the empty set keeps the set unchanged.
• A ∩ U = A: Intersection with the universal set returns the original set.

6. Complement Laws

• A ∪ A^c = U: The union of a set with its complement results in the universal set.
• A ∩ A^c = ∅: A set and its complement share no elements in common.

7. De Morgan’s Laws

• (A ∪ B)^c = A^c ∩ B^c: The complement of a union equals the intersection of the complements.
• (A ∩ B)^c = A^c ∪ B^c: The complement of an intersection equals the union of the complements.

8. Absorptive Laws

• A ∪ (A ∩ B) = A: When a set is united with its intersection with another set, the result is the original set.
• A ∩ (A ∪ B) = A: A set absorbs the union with another set.

These fundamental laws make it easier to manipulate and simplify set expressions. They are widely used in logic, mathematics, computer science, and related fields.