## Thinking Mathematically (6th Edition)

First, perform the operation inside the parentheses of the set$A\cap \left( B\cup C \right)$. Now, compute$B\cup C$. Set $B\cup C$ contains all the elements thatare in set B,set C,or in both. 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 B and set Care II, III, IV, VI,and VII,so it represents the set$B\cup C$. Now, find the intersection of the set $B\cup C$ and set A. Common regions of both the sets together represent the set$A\cap \left( B\cup C \right)$. So, regions II, IV, and V represent the set$A\cap \left( B\cup C \right)$. Perform the operation inside the parentheses of the set$\left( A\cap B \right)\cup C$. Now, compute$A\cap B$. Set $A\cap B$ contains all the common elements of the set A and set B. In the provided Venn diagram,common regions of set A and set Bare II and V,so together they represent the set$A\cap B$. Now,find the union of the set $A\cap B$ and set C. In the Venn diagram, union of regions of both the sets are II, IV, V, VI, and VII,and they together represent the set$\left( A\cap B \right)\cup C$. Therefore, set $A\cap \left( B\cup C \right)$is represented by regions II, IV, and V and set $\left( A\cap B \right)\cup C$is represented by regions II, IV, V, VI, and VII. Both the sets are represented by different regions, so they are not equal.