Question:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) $(A \cup B)^{\prime}=A^{\prime} \cap B$,
(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$ '.
Solution:
Given:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
We have to verify:
(i) $(A \cup B)^{\prime}=A^{\prime} \cap B$,
LHS
$A \cup B=\{2,3,4,5,6,7,8\}$
$(A \cup B)^{\prime}=\{1,9\}$
RHS
$A^{\prime}=\{1,3,5,7,9\}$
$B^{\prime}=\{1,4,6,8,9\}$
$A^{\prime} \cap B^{\prime}=\{1,9\}$
LHS = RHS
Hence proved.
(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$
LHS
$A \cap B=\{2\}$
$(A \cap B)^{\prime}=\{1,3,4,5,6,7,8,9\}$
RHS
$A^{\prime}=\{1,3,5,7,9\}$
$B^{\prime}=\{1,4,6,8,9\}$
$A^{\prime} \cup B^{\prime}=\{1,3,4,5,6,7,8,9\}$
LHS = RHS
Hence proved.