## Thinking Mathematically (6th Edition)

(a) First, perform the operation inside the parentheses of the set$\left( A\cap B \right)\cup C$. Now, we compute$A\cap B$. Set $A\cap B$ contains all the elements thatare common to both sets$A$and$B$. In the provided Venn diagram,regions I, II, IV, and V represent the set Aregions II, III, V, and VI represent the set B. The common regions of A and B are II and V that represent the set$A\cap B$. Now, we willfind the union of the set $A\cap B$ and set C. Union of the regions of both the sets represents the set$\left( A\cap B \right)\cup C$. So, regions II, IV, V, VI, and VII represent the set$\left( A\cap B \right)\cup C$. (b) First, perform the operation inside the parentheses of the set$\left( A\cup C \right)\cap \left( B\cup C \right)$. In the provided Venn diagram,regions I, II, IV, and V represent the set A,regions II, III, V, and VI represent the set B,andregions IV, VI, VI, and VII represent the set C. Now, the union of regions of A and C are I, II, IV, V, VI ,and VII that represent the set$A\cup C$. Similarly, find set$B\cup C$. Union of the regions of both the sets B and C together represents the set$B\cup C$. Now, the common regions of the set $A\cup C$and set$B\cup C$ are II, IV, V, VI, and VII. Together they represent the set$\left( A\cup C \right)\cap \left( B\cup C \right)$. (c) In parts (a) and (b)set$\left( A\cap B \right)\cup C$ and set$\left( A\cup C \right)\cap \left( B\cup C \right)$,respectively, represent the same regions II, IV, V, VI, and VII. Both the sets are represented by same regions. So, they are equal.