## Thinking Mathematically (6th Edition)

(a) First, perform the operation inside the parentheses of the set$C\cup \left( B\cap A \right)$. Now, compute$B\cap A$. Set $B\cap A$ contains all the elements thatare common to both set B and set A. 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 A are II and V,so it represents the set$B\cap A$. Now, find the union of the set $B\cap A$ and set C. Union of regions of both the sets together represent the set$C\cup \left( B\cap A \right)$. So, regions II, IV, V, VI, and VII represent the set$C\cup \left( B\cap A \right)$. (b) First, perform the operation inside the parentheses of the set$C\cap \left( B\cup A \right)$. Now, compute$B\cup A$. Set $B\cup A$ contains all the elements thatare in set B,set A,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, andregions I, II, IV, and V represent the set A. The union of regions of set B and set Aare I, II, III, IV, V, and VI, and together they represent the set$B\cup A$. Now, find the intersection of the set $B\cup A$ and set C. Common regions of both the sets together represent the set$C\cap \left( B\cup A \right)$. So, regions IV, V, and VI represent the set$C\cap \left( B\cup A \right)$. (c) In part (a), set $C\cup \left( B\cap A \right)$is represented by the regions II, IV, V, VI, and VII,and in part (b), set $C\cap \left( B\cup A \right)$is represented by the regions IV, V, and VI. Both the sets are represented by different regions. Therefore, they are not equal and, hence, they may differ.