# DEMORGANS LAW

Demorgans law :

De Morgan’s laws relate the three basic set operations Union, Intersection and Complementation.

## De morgan's laws

De morgan's law for set difference :

For any three sets A, B and C, we have

(i)  A \ (B u C)  =  (A \ B) n (A \ C)

(ii)  A \ (B n C)  =  (A \ B) u (A \ C)

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 \ (B n C)  =  (A \ B) u (A \ C)

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

A \ (B n C)  =  (A \ B) u (A \ C)

Hence, De morgan's law for set difference is verified.

Now, let us look at the Venn diagram proof of De morgan's law for complementation.

(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.

## De morgan's laws - Practice problems

Let us look at some practice problems on "Demorgans law"

Problem 1 :

Let A  =  { a, b, c, d, e, f, g, x, y, z }, B  =  { 1, 2, c, d, e } and

C  =  { d, e, f, g, 2, y }. Verify De Morgan’s laws of set difference.

Solution :

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

To do this, we consider

B u C  =  { 1, 2, c, d, e } u { d, e, f, g, 2, y }

B u C  =  { 1, 2, c, d, e, f, g, y }

We know that

A / (B u C)  =  { a, b, c, d, e, f, g, x, y, z }  \ { 1, 2, c, d, e, f, g, y }

A / (B u C)  =  { a, b, x, z } ---------(1)

A \ B   =  { a, b, f, g, x, y, z }

A \ C   =  { a, b, c, x, z }

(A \ B) n (A \ C)  =  { a, b, x, z } ---------(2)

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

Similarly, one can verify A \ (B n C)  =  (A \ B) u (A \ C) for the given sets above.

Let us look at the next problem on "Demorgans law"

Problem 2 :

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' n B'  =   { -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'

Similarly, one can verify (A n B)'  =  A' u B' for the given sets above.

After having gone through the stuff given above, we hope that the students would have understood "Demorgans law".

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