Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:
(i) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$
(ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
(iii) $A \cap(B-C)=(A \cap B)-(A \cap C)$
(iv) $A-(B \cup C)=A(A-B) \cap(A-C)$
(v) $A-(B \cap C)=(A-B) \cup(A-C)$
(vi) $A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C)$.
Given:
A = {1, 2, 4, 5}, B = {2, 3, 5, 6} and C = {4, 5, 6, 7}
We have to verify the following identities:
(i) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$
LHS
$(B \cap C)=\{5,6\}$
$A \cup(B \cap C)=\{1,2,4,5,6\}$
RHS
$(A \cup B)=\{1,2,3,4,5,6\}$
$(A \cup C)=\{1,2,4,5,6,7\}$
$(A \cup B) \cap(A \cup C)=\{1,2,4,5,6\}$
LHS = RHS
$\therefore A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$
(ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
LHS
$(B \cup C)=\{2,3,4,5,6,7\}$
$A \cap(B \cup C)=\{2,4,5\}$
RHS
$A \cap B=\{2,5\}$
$A \cap C=\{4,5\}$
$(A \cap B) \cup(A \cap C)=\{2,4,5\}$
LHS = RHS
$\therefore A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
(iii) $A \cap(B-C)=(A \cap B)-(A \cap C)$
LHS
$(B-C)=\{2,3\}$
$A \cap(B-C)=\{2\}$
RHS
$(A \cap B)=\{2,5\}$
$(A \cap C)=\{4,5\}$
$(A \cap B)-(A \cap C)=\{2\}$
LHS = RHS
$\therefore A \cap(B-C)=(A \cap B)-(A \cap C)$
(iv) $A-(B \cup C)=(A-B) \cap(A-C)$
LHS
$(B \cup C)=\{2,3,4,5,6,7\}$
$A-(B \cup C)=\{1\}$
RHS
$(A-B)=\{1,4\}$
$(A-C)=\{1,2\}$
$(A-B) \cap(A-C)=\{1\}$
LHS = RHS
$\therefore A-(B \cup C)=(A-B) \cap(A-C)$
(v) $A-(B \cap C)=(A-B) \cup(A-C)$
LHS
$(B \cap C)=\{5,6\}$
$A-(B \cap C)=\{1,2,4\}$
RHS
$(A-B)=\{1,4\}$
$(A-C)=\{1,2\}$
$(A-B) \cup(A-C)=\{1,2,4\}$
LHS = RHS
$\therefore A-(B \cap C)=(A-B) \cup(A-C)$
(vi) $A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C)$
LHS
$(B \Delta C)=(B-C) \cup(C-B)$
$(B-C)=\{2,3\}$
$(C-B)=\{4,7\}$
$(B-C) \cup(C-B)=\{2,3,4,7\}$
$\Rightarrow(B \Delta C)=\{2,3,4,7\}$
$A \cap(B \Delta C)=\{2,4\}$
RHS
$(A \cap B)=\{2,5\}$
$(A \cap C)=\{4,5\}$
$(A \cap B) \Delta(A \cap C)=\{(A \cap B)-(A \cap C)\} \cup\{(A \cap C)-(A \cap B)\}$
$(A \cap B)-(A \cap C)=\{2\}$
$(A \cap C)-(A \cap B)=\{4\}$
$\{(A \cap B)-(A \cap C)\} \cup\{(A \cap C)-(A \cap B)\}=\{2,4\}$
$\Rightarrow(A \cap B) \Delta(A \cap C)=\{2,4\}$
LHS = RHS
$\therefore A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C)$
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