Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},

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.