DE MORGAN'S LAWS

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

De Morgan's First Law :

The complement of the union of two sets  is equal to the intersection of their complements. 

That is

(AuB)' = A'nB'

De Morgan's Second Law :

The complement of the intersection of two sets is equal to the union of their complements. 

That is, 

(AnB)' = A'uB'

De Morgan's First Law - Proof by Venn diagram

(AuB)' = A'nB'

Let's draw Venn diagram for (AuB)'. 

To draw Venn diagram for (AuB)', first draw Venn diagram for (AuB).

To draw Venn diagram for (AuB)', shade the region other than (AuB).

Now, let's draw Venn diagram for (A'n B'). 

Draw Venn diagram for A' and B'.

Then, to draw Venn diagram for (A'nB'), find the common region of A' and B'. 

The resulting Venn diagrams of (AuB)' and (A'nB') are same. 

So, 

(AuB)' = A'nB'

De Morgan's First Law - Proof by Venn diagram

(AnB)' = A'uB'

Let's draw Venn diagram for (AnB). 

To draw Venn diagram for (AnB)', first draw Venn diagram for (AnB)

To draw Venn diagram for (AnB)', shade the region other than (AnB). 

Now, let's draw Venn diagram for (A'u B').

Draw Venn diagram for A' and B'.

Then, to draw Venn diagram for (A'uB'), join A' and B'. 

The resulting Venn diagrams of (AnB)' and (A'uB') are same. 

So, 

(AnB)' = A'uB'

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