(11) For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that
(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)
(iii) Verify using Venn diagrams
Solution:
(i) A U (B ∩ C) = (A U B) ∩ (A U C)
L.H.S
A U (B ∩ C)
(B ∩ C) = {-1,-2,3,4,5,6} ∩ {-1,2,3,4,5,7}
= {-1,3,4,5}
A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}
= {-3,-1,0,3,4,5,6,8,10} --- (1)
R.H.S
(A U B) ∩ (A U C)
(A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}
= {-3,-2,-1,0,3,4,5,6,8,10}
(A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}
= {-3,-1,0,2,3,4,5,6,7,8,10}
(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}
= {-3,-1,0,3,4,5,6,8,10} --- (2)
(1) = (2)
A U (B ∩ C) = (A U B) ∩ (A U C)
Hence proved
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)
A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}
L.H.S
(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}
= {-2,-1,2,3,4,5,6,7}
A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}
= {-1,4,6} ---- (1)
R.H.S
(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}
= {-1,4,6}
(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}
= {-1,4}
(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}
= {-1,4,6} ---- (2)
(1) = (2)
A ∩ (B U C) = (A ∩ B) U (A ∩ C)
Hence proved