Chapter 2 - Set Theory - 2.4 Set Operations and Venn Diagrams with Three Sets - Exercise Set 2.4 - Page 93: 67

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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,regions I, II, IV, and V represent the set A,and regions I, II, III, IV, V, VI, VII, and VIII represent the set U. The union of regions of set B and set C are II, III, IV, V, VI, and VII,so it represents the set\[B\cup C\],and its complement is represented by the regions I and VIII whichafter removing regions of the set \[B\cup C\]from the universal set U(represented by all the regions I, II, III, IV, V, VI, VII and VIII) gives the set \[\left( B\cup C \right)'\]. Now, find the intersection of the set \[\left( B\cup C \right)'\] and set A. Common regions of both the sets together represent the set\[A\cap \left( B\cup C \right)'\]. So, region I represents the set\[A\cap \left( B\cup C \right)'\]. Perform the operation inside the parentheses of the set\[A\cap \left( B'\cap C' \right)\]. Now, compute\[B'\cap C'\]. For this, complement of the set Bis represented by the regions I, IV, VII, and VIII thatcame after removing regions of set Bfrom the universal set Uand complement of the set C is represented by the regions I, II, III, and VIII thatcame after removing regions of set C from the universal set U. Set\[B'\cap C'\]contains all the elements that are common elements of sets\[B'\]and\[C'\]. In the provided Venn diagram,common regions of set \[B'\]and set \[C'\]are I and VIII,so together they represent the set\[B'\cap C'\]. Now,find the intersection of the set \[B'\cap C'\] and set A. In the Venn diagram, common regions of both the sets is I. They together represent the set\[A\cap \left( B'\cap C' \right)\]. Therefore, set \[A\cap \left( B\cup C \right)'\]is represented by region I and set \[A\cap \left( B'\cap C' \right)\]is also represented by regionI. Both the sets are represented by same region, so they are equal.