I.BCOM

                             UNIT–1 

                                            DE-MORGANS LAW

De Morgan’s Laws:

For any two sets A and B, 

(A∪B)

|=A|∩B| and 

(A∩B)

|=A

|∪B|

For any three sets A, B, and C.

A-(B∪C)=(A-B)∩(A-C)

A-(B∩C)=(A-B)∪(A-C).

1. Verify the distributive law 

A∪(B∩C)=(A∪B)∩(A∪C) 

 Example 1

If A={1,2,3,4}, B={2,4,5,6} and C={1,3,5}, verify 

that A∩(B∪C)=(A∩B)∪(A∩C).

Proof:

Given that A={1,2,3,4}, B={2,4,5,6} and C={1,3,5}

Consider, LHS A∩(B∪C)

(B∪C)={2,4,5,6}∪{1,3,5}

 ={1,2,3,4,5,6}

A∩(B∪C)={1,2,3,4}…………(1)

(A∩B)= {1,2,3,4}∩{2,4,5,6}={2,4}

(A∩C)= {1,3}

(A∩B)∪(A∩C)= {1,2,3,4}………(2)

A∩(B∪C)=(A∩B)∪(A∩C)

Hence Proved.

(ii) 

 A-(B∩C)=(A-B)∪(A-C).

Consider, A-(B∩C)

(B∩C)={2,3,4,5,6}∩{4,5,6,7,8,9}

 ={4,5,6}

A-(B∩C)={0,1,3,4,6,7,9,10}-{4,5,6}

 ={0,1,3,7,9,10}………….(3)

(A-B)={0,1,3,4,6,7,9,10}-{2,3,4,5,6}

={0,1,7,9,10}

(A-C)={0,1,3,4,6,7,9,10}-{4,5,6,7,8,9}

={0,1,3,10}

(A-B)∪(A-C)={0,1,7,9,10}∪{0,1,3,10}

 ={0,1,3,7,9,10}……………..(4)

From (3) and (4), A-(B∩C)=(A-B)∪(A-C)...

     

                             Thank you 

 I.BCOM 

A.P MUKHIL 

23UCM026