VERIFYING DEMORGAN'S LAWS

Question 1 :

If A = {–2, 0, 1, 3, 5}, B = {–1, 0, 2, 5, 6} and C = {–1, 2, 5, 6, 7}, then show that A − (B U C) = (A − B) n (A − C)

Solution :

(B U C)  =  {-1, 0, 2, 5, 6, 7}

A − (B U C)  =  {-2, 1, 3} ----(1)

(A − B)  =  {-2, 1, 3}

(A − C)  =  {-2, 0, 1, 3}

 (A − B) n (A − C)  =  {-2, 1, 3} ----(2)

(1)  =  (2)

Hence proved.

Question 2 :

If A = {y : y = (a + 1)/2, a ∈ W and a ≤ 5}, B  =  {y : y = (2n - 1)/2, n ∈ W and n < 5} and C  =  {-1, -1/2,1, 3/2, 2}, then show that A - (BUC)  =  (A - B) n (A - C)

Solution :

A = {y : y = (a + 1)/2, a ∈ W and a ≤ 5}

A  = {1/2, 3/2, 2, 5/2, 3}

B  =  {y : y = (2n - 1)/2, n ∈ W and n < 5}

B = {-1/2, 1/2, 3/2, 5/2, 7/2}

C  =  {-1, -1/2, 1, 3/2, 2}

A - (BUC)  =  (A - B) n (A - C)

B U C  =  {-1/2, -1, 1/2, 1, 3/2, 2, 5/2, 7/2}

A - (BU C)  =  {3}  -----(1)

(A - B)  =  {2, 3}

(A - C)  =  {1/2, 5/2, 3}

 (A - B) n (A - C)  =  {3}  -----(2)

(1)  =  (2)

Hence proved.

Question 3 :

Verify A − (B n C) = (A − B) U (A − C) using Venn diagrams.

Solution :

Question 4 :

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.

Solution :

De morgan's laws for complementation :

(i)  (A U B)'  =  A' n B'

(ii)  (A n B)'  =  A' U B'

 A = {7, 8, 11, 12} and B = {4, 8, 12, 15},

 U = {4, 7, 8, 10, 11, 12, 15, 16}

(i) 

(A U B)  =  {4, 7, 8, 11, 12, 15}

(A U B)'  =  {10, 16} ----(1)

A'  =  {4, 10, 15, 16}

B'  =  {7, 10, 11, 16}

A' n B'  =  {10, 16}  ----(2)

Hence proved.

(ii) 

(A n B)  =  {8, 12}

(A n B)'  =  {4, 7, 10, 11, 15, 16} ----(1)

A'  =  {4, 10, 15, 16}

B'  =  {7, 10, 11, 16}

A' U B'  =  {4, 7, 10, 11, 15, 16}  ----(2)

Hence proved.

Question 5 :

Verify (A n B)' = A' U B' using Venn diagrams.

Solution :

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