(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