Prove DeMorgans laws a A B U C A B U A C b A B n C A

Prove DeMorgan\'s laws:

a) A - (B U C) = ( A - B) U (A - C).

b) A - (B n C) = ( A - B) n (A - C).

Solution

Prove DeMorgan\'s laws:

a) A - (B U C) = ( A - B) n (A - C).

b) A - (B n C) = ( A - B) U (A - C).

let A = {1,2,3,4,5}, B = {3,4,5,6,7}, C = { 6,7,8}

so BUC = {3,4,5,6,7} U { 6,7,8} = {3,4,5,6,7,8,9}

A - BUC = {1,2,3,4,5} - {3,4,5,6,7,8,9} = {1,2}

A - B = {1,2,3,4,5} -{3,4,5,6,7} = {1,2}

A - C = {1,2,3,4,5} - { 6,7,8} = {1,2,3,4,5}

( A - B) n (A - C) = {1,2} U {1,2,3,4,5} = {1,2}

Hence we proved A - (B U C) = ( A - B) n (A - C)

Then

B n C = {3,4,5,6,7} n { 6,7,8} = {6,7}

A - (B n C) = {1,2,3,4,5} - {6,7} = {1,2,3,4,5}

A - B = {1,2,3,4,5} -{3,4,5,6,7} = {1,2}

A - C = {1,2,3,4,5} - { 6,7,8} = {1,2,3,4,5}

( A - B) U (A - C) = {1,2} U {1,2,3,4,5} = {1,2,3,4,5}

Hence we proved A - (B n C) = ( A - B) U (A - C)