DE MORGANS LAW FOR COMPLIMENTS

De Morgan’s father (a British national) was in the service of East India Company, India. Augustus De Morgan (1806-1871) was born in Madurai, Tamilnadu, India. His family moved to England when he was seven months old. He had his education at Trinity college, Cambridge, England.

De Morgan’s laws relate the three basic set operations union, intersection and complement.

De morgan's law for set complementation :

Let U be the universal set containing sets A and B. Then

(i)  (A u B)'  =  A' n B'

(ii)  (A n B)'  =  A' u B'

Proof by Venn diagram

(A n B)'  =  A' u B'

From the above Venn diagrams (2) and (5), it is clear that 

(A n B)'  =  A' u B'

Hence, De morgan's law for complementation is verified.

(A U B)'  =  A' n B'

Solved Examples

Example 1 :

Let U  =  {-2, -1, 0, 1, 2, 3, 4, 5, .......10 }, A = {-2, 2, 3, 4, 5} and B = {1, 3, 5, 8, 9}. Verify De Morgan’s laws of complementation

Solution : 

First, we shall verify (A u B)'  =  A' n B'

To do this, we consider

A u B  =  {-2, 2, 3, 4, 5} u {1, 3, 5, 8, 9}

A u B  =  {-2, 1, 2, 3, 4, 5, 8, 9}

We know that

(A u B)'  =  U \ {-2, 1, 2, 3, 4, 5, 8, 9}

(A u B)'  =  {-1, 0, 6, 7, 10} -----(1)

A'  =  U \ A   =  U \ { -2, 2, 3, 4, 5 }

=  {-1, 0, 1, 6, 7, 8, 9, 10}

B'  =  U \ B  =  U \ {1, 3, 5, 8, 9}

=  {-2, -1, 0, 2, 4, 6, 7, 10}

A'nB' =  {-1, 0, 1, 6, 7, 8, 9, 10} n {-2, -1, 0, 2, 4, 6, 7, 10}

A' n B'  =  {-1, 0, 6, 7, 10} -----(2)

From (1) and (2), it is clear that (A u B)'  =  A' n B'

First, we shall verify (A n B)'  =  A'u B'.

To do this, we consider

A n B  =  {- 2, 2, 3, 4, 5 } u {1, 3, 5, 8, 9}

A n B  =  {3, 5}

We know that

(A n B)'  =  U \ { 3, 5 }

(A n B)'  =  {2, -1, 0, 1, 2, 4,  6, 7, 8, 9, 10}

A'  =  U \ A   =  U \ {-2, 2, 3, 4, 5}

   =  {-1, 0, 1, 6, 7, 8, 9, 10}

B'  =  U \ B  =  U \ {1, 3, 5, 8, 9}

=  {-2, -1, 0, 2, 4, 6, 7, 10}

A'UB' =  {-1, 0, 1, 6, 7, 8, 9, 10} n {-2, -1, 0, 2, 4, 6, 7, 10}

A' U B'  =  {2, -1, 0, 1, 2, 4,  6, 7, 8, 9, 10} -----(2)

From (1) and (2), it is clear that

(A n B)'  =  A' U B'

Example 2 :

Let U = {4, 8, 12, 16, 20, 24, 28} , A = {8, 16, 24} and B = {4, 16, 20, 28}. Find (AU B)' and (A n B)'.

Solution : 

A u B  =  {8, 16, 24} U {4, 16, 20, 28}

A u B  =  { 4, 8, 16, 20, 24, 28 }

We know that

(A u B)'  =  U \ { 4, 8, 16, 20, 24, 28 }

(A u B)'  =  { 12 }

(A n B)  =  {8, 16, 24} n {4, 16, 20, 28}

  =  {16}

(A n B)'  =   U \ { 16 }

  =  {4, 8, 12, 20, 24, 28}

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