Absorption Law

The statement of absorption law - The absorption law states that

Law 1 - ORing of a boolean variable A with the AND of boolean variables A & B is equal to that boolean variable A itself.

i.e.,           A  + A . B  =  A

Law 2 - ANDing of a boolean variable A with the OR of boolean variables A & B is equal to that boolean variable A itself.

i.e.,          A  . ( A  + B)  =  A
Proof of Absorption Law Using Truth Table

We are proving both the laws using truth table method as follows -

To Prove law 1  :    A  + A . B  =  A

Proof  :
A B A . B A + A . B
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1

By referring to columns 1 & 4 we observe that  A + A . B = A

To Prove law2  :    A  . ( A + B )  =  A

Proof  :
A B A + B A .( A + B )
0 0 0 0
0 1 1 0
1 0 1 1
1 1 1 1

By referring to columns 1 & 4 we observe that  A .( A + B)  = A

Hence both the absorption laws are proved using truth table.
Proof of Absorption Law Using Algebraic Method

The algebraic method uses boolean laws to prove the absorption laws -

To prove :           A + A . B = A

Proof :

LHS = A + A . B

= A . 1  + A . B   [ using identity law A .1 = A ]

= A . ( 1 + B )    [ using distributive law A . ( B + C ) = A . B  + A . C ]

=  A . 1              [ using identity law 1 + B = 1 ]

= A                   [ using identity law A . 1 = A]

= RHS

Hence absorption  law 1    A + A . B  =  A is proved.

To prove :       A . ( A + B ) = A

Proof -

LHS = A . ( A + B )

= ( A + 0 )  + ( A +B ) [ using null law A + 0 = A ]

= A + ( 0 . B )    [ using distributive law A + ( B . C ) = ( A + B  ).( A + C ) ]

=  A + 0             [ using null law 0 . B = 0 ]

= A                   [ using null law A + 0 = A ]

= RHS

Hence absorption law 2    A .( A + B )  =  A is proved.