Set Theory: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)


To verify distributive law for three given non-empty sets A, B and C, that is, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Distributive Law

  • Distributive Law states that, the sum and product remain the same value even when the order of the elements is altered.
  • We will see the distributive property of sets using Venn diagram. Before start with the distributive property. Lets see some basic concepts of set theory. 
  • Distributive laws for three given non-empty sets A, B, and C, that is, A∪(B∩C) = (A∪B)∩(A∪C)

     Venn Diagram

        A Venn diagram in math is used in logic theory and set theory to show various sets of data and their relationship with each other.

     Set Theory

  • Set Theory is a branch of mathematical logic where we learn sets and their properties. A set is a collection of well defined objects or groups of objects. 
  • These objects are often called elements or members of a set. For example, a group of players in a cricket team is a set. 

Prerequisite Knowledge

Null Set

In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.

Non-Empty Set 

  • A nonempty set is a set containing one or more elements. Any set other than the empty set is called non-empty set. Nonempty sets are sometimes also called nonvoid sets.
  • A nonempty set containing a single element is called a singleton set.


The union of two sets A and B is the set of all those elements which are either in A or B, i.e. A ∪ B


The intersection of two sets A and B is the set of all common elements. The intersection of these two sets is denoted by A∩B.

Representation of LHS relations of distributive law of set theory

Representation of RHS relations of distributive law of set theory