## Thinking Mathematically (6th Edition)

First, perform the operation inside the parentheses of the set$B\cap \left( A\cup C \right)$. Now, compute$A\cup C$. Set $A\cup C$ contains all the common elements of set A and set C. In the provided Venn diagram,regions IV, V, VI, and VII represent the set C, regions II, III, V, and VI represent the set B, and regions I, II, IV, and V represent the set A. The union of regions of set A and set C are I, II, IV, V, VI, and VII, so it represents the set$A\cup C$. To find the intersection of the set $A\cup C$ and set B, take common regions of both the sets, which together represent the set$B\cap \left( A\cup C \right)$. So, regions II, V, and VI represent the set$B\cap \left( A\cup C \right)$. Perform the operation inside the parentheses of the set$\left( A\cap B \right)\cup \left( B\cap C \right)$. Now, compute$A\cap B$. Set $A\cap B$ contains all the common elements of set Aand set B. In the provided Venn diagram, common regions of set A and set B are II and V,so together they represent the set$A\cap B$. Then, compute$B\cap C$. Set $B\cap C$ contains all the common elements of the set Band set C. In the provided Venn diagram,common regions of set B and set C are II, V, and VI, so together they represent the set$B\cap C$. Now, find the union of the set $A\cap B$ and set$B\cap C$. In the Venn diagram, take union of all the regions of both sets that are II, V, and VI, they together represent the set$\left( A\cap B \right)\cup \left( B\cap C \right)$. Therefore, set $B\cap \left( A\cup C \right)$ and set$\left( A\cap B \right)\cup \left( B\cap C \right)$are represented by same regions II, V, and VIand, hence,they are equal.