Question

10. If U = {4,7,8,10,11,12,15,16), A={7,8,11,12,} and B = {4,8,12,15}, then verify
De Morgan's Laws for complementation. this answer​

Answer 1

We know that De Morgan's Laws for complementation is

$(A \cup \: B)^{'} = {A}^{'} \cap \: {B}^{'} \\ \\ (A \cap \: B)^{'} = {A}^{'} \cup \: {B}^{'}$
Find
$A \cup \: B = [4 ,\: 7, \: 8, \: 11, \: 12, \: 15] \\ \\ (A \cup \: B)^{'} = [10 \:, 16].....eq1 \\ \\ {A}^{'} = [4 \:, 10 \: ,15 \:, 16]\\ \\ { B}^{'} = [7 \:, 10 \:, 11 \:, 16] \\ \\ {A}^{'} \cap \: {B}^{'} = [10 \:, 16]...eq2 \\ \\ eq1 = eq2$
De Morgan's Laws first law is proved.

For second law :

$(A \cap \: B)^{'} = {A}^{'} \cup \: {B}^{'} \\ \\ (A \cap \: B) = [8 \:, 12]\\ \\ (A \cap \: B)^{'} = [4 \:, 7 \: ,10 \:, 11 \:, 15 \:, 16]\\ \\ {A}^{'} = [4 \:, 10 \:, 15 \:, 16] \\ \\ {B}^{'} = [7 \:, 10 \: ,11 \:, 16] \\ \\ {A}^{'} \cup \: {B}^{'} = [4 \:, 7 \:, 10 \:, 11 \:, 15 \:, 16] \\ \\ LHS=RHS$
Hence proved