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}

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Resources (Videos, Concepts, Worksheets and More)

    Jul 11, 24 05:20 PM

    SAT Math Resources (Videos, Concepts, Worksheets and More)

    Read More

  2. SAT Math Videos (Part - 21)

    Jul 11, 24 05:14 PM

    satmath18.png
    SAT Math Videos (Part - 21)

    Read More

  3. Best Way to Learn Mathematics

    Jul 11, 24 11:59 AM

    Best Way to Learn Mathematics

    Read More