Associative and Distributive Laws of Set Operations

Associative Law

Associative laws establish the rules of taking unions and intersections of sets. They are applicable to all sets including the set of real numbers.

𝐴 𝑈 (𝐵 𝑈 𝐶) = (𝐴 𝑈 𝐵) 𝑈 𝐶

This law states that by taking the union of a set to the union of two other sets is the same as taking union of the original set and one of the other two sets, and then by taking the union of the results with the last set. 

𝐴 ∩(𝐵 ∩𝐶) = (𝐴 ∩𝐵) ∩𝐶

This law states that taking the intersection of a set to the intersection of two other sets is the same as taking the intersection of the original set and one of the other two sets, and then taking the intersection of the results with the last set.

Distributive Law

Distributive laws also establish the rules of taking unions and intersections of sets.

 𝐴 𝑈 (𝐵 ∩𝐶) = (𝐴 𝑈 𝐵) ∩(𝐴 𝑈 𝐶)

This law states that by taking the union of a set to the intersection of two other sets is the same as taking the union of the original set and both the other two sets separately, and then taking the intersection of the results. 

𝐴 ∩(𝐵 𝑈 𝐶) = (𝐴 ∩𝐵) 𝑈 (𝐴 ∩𝐶)

This law states that taking the intersection of a set to the union of two other sets is the same as taking the intersection of the original set and both the other two sets separately, and then taking the union of the results.