Chapter 2 - Set Theory - 2.4 Set Operations and Venn Diagrams with Three Sets - Exercise Set 2.4 - Page 93: 65

True

First, perform the operation inside the parentheses of the set\[B\cup \left( A\cap C \right)\]. Now, compute\[A\cap C\]. Set \[A\cap C\] contains all the common elements of set A and 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, andregions I, II, IV, and V represent the set A. Now, the common regions of set A and set C are IV and V, so it represents the set\[A\cap C\]. Now,find the union of the set \[A\cap C\] and set B. Take union of all regions of both the sets, which together represent the set\[B\cup \left( A\cap C \right)\]. So, regions II, III, IV, V, and VI represent the set\[B\cup \left( A\cap C \right)\]. Perform the operation inside the parentheses of the set\[\left( A\cup B \right)\cap \left( B\cup C \right)\]. Now, compute\[A\cup B\]. Set \[A\cup B\] contains all the elements thatare in the set A,set B, or in both. In the provided Venn diagram,take union of the regions of set A and set B asI, II, III, IV, V, VI, and V,which together represent the set\[A\cup B\]. Then, compute\[B\cup C\]. Set \[B\cup C\] contains all the elements thatare in the set B,set C, or in both. In the provided Venn diagram,Take union of the regions of set B and set C thatare II, III, IV, V, VI, and VII,whichtogether they represent the set\[B\cup C\]. Now,find the intersection of the set \[A\cup B\] and set\[B\cup C\]. In the Venn diagram,common regions of both sets are II, III, IV, V, and VI, and they together represent the set\[\left( A\cup B \right)\cap \left( B\cup C \right)\]. Therefore, set \[B\cup \left( A\cap C \right)\] and set\[\left( A\cup B \right)\cap \left( B\cup C \right)\]are represented by same regions II, III, IV, V, and VI,so they are equal.