Chapter 2 - Set Theory - 2.4 Set Operations and Venn Diagrams with Three Sets - Exercise Set 2.4 - Page 93: 68

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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 elements thatare common to theset Band 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,regions I, II, IV, and V represent the set A, andregions I, II, III, IV, V, VI, VII, and VIII represent the set U. Now, the common regions of set B and set C are V and VI, so it represents the set\[B\cap C\],and its complement is represented by the regions I, II, III, IV,VII, and VIII which after removing regions of the set \[B\cap C\]from the universal set U(represented by all the regions I, II, III, IV, V, VI, VII and VIII) gives the set \[\left( B\cap C \right)'\]. Now, find the union of the set \[\left( B\cap C \right)'\] and set A. Union of the regions of both the sets together represent the set\[A\cup \left( B\cap C \right)'\]. So, regionsI, II, III, IV, V, VII, and VIII represent the set\[A\cup \left( B\cap C \right)'\]. Perform the operation inside the parentheses of the set\[A\cup \left( B'\cup C' \right)\]. Now, compute\[B'\cup C'\]. 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 U,and 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\[B'\cup C'\]contains all the elements that are in set \[B'\],set\[C'\], or in both. In the provided Venn diagram,union of all the regions of set \[B'\]and set \[C'\]are I, II, III, IV, VII, and VIII,so together they represent the set\[B'\cup C'\]. Now, find the union of the set \[B'\cup C'\] and set A. In the Venn diagram, union of all regions of both the sets are I, II, III, IV, V, VII, and VIII. They together represent the set\[A\cup \left( B'\cup C' \right)\]. Therefore, set \[A\cup \left( B\cap C \right)'\]is represented by regions I, II, and IV and set \[A\cup \left( B'\cup C' \right)\]is represented by regions I, II, III, IV, V, VII, and VIII. Both the sets are represented by different regions, so they are not equal.