laws and properties of sets

I B. COM 

                   UNIT 1

                          Laws and Properties of Sets:

Sets have the following properties:

1. Commutative Laws:

For any two sets A and B, 

A∪B=B∪A

A∩B=B∩A. But A-B≄B-A

2. Associative Laws:

For any three sets A, B, and C.

A∪(B∪C)=(A∪B)∪C

A∩(B∩C)=(A∩B)∩C

3. Distributive Laws:

For any three sets A, B, and C.

A∪(B∩C)=(A∪B)∩(A∪C) 

A∩(B∪C)=4. 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).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.

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                   Thank you 

D. GOWTHAM

23UCM011