## Thinking Mathematically (6th Edition)

First, perform the operation inside the parenthesis of the set$\left( A\cup B \right)'$. So we compute$A\cup B$. Set $A\cup B$ contains all the elements which are either in set A or set B or in both. In the Venn diagram, Regions II, III, V and VI represent the set B. Regions I, II, IV and V represent the set A. Now the union of regions of set A and set B are I, II, III, IV, V and VI. So it represents the set$A\cup B$. To find the complement of the set$A\cup B$, it contains all the elements of the universal set Uexcept the elements of set$A\cup B$. So region VII and VIII represent the set$\left( A\cup B \right)'$. Then common region of both the sets$\left( A\cup B \right)'$and C is region VII only. So, region VII represents the set $\left( A\cup B \right)'\cap C$ Observe that it involved only set C. since region VII contains element of C only.