## Thinking Mathematically (6th Edition)

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