Absorption Laws

Date: 04/10/2001 at 20:44:44
From: Tony Sorapuru
Subject: Boolean rule

   x+xy=x+y

Date: 04/11/2001 at 14:08:35
From: Doctor Twe
Subject: Re: Boolean rule

Hi Tony - thanks for writing to Dr. Math.

I think perhaps you mean x + x'y = x + y, or maybe x + xy = x. These 
are known as the "absorption laws" [along with x(x'+y) = xy and 
x(x+y) = x]. They can be proved by truth table or by replacement. For 
example:

      x y | x'y | x+x'y   x+y
     -----+-----+------------
      0 0 |  0  |   0      0
      0 1 |  1  |   1      1
      1 0 |  0  |   1      1
      1 1 |  0  |   1      1

Because the final two columns of the truth table are identical, we can 
say that the two expressions, x + x'y and x + y, are equivalent. We 
can also show their equivalence by replacement:

       x+x'y
     = (x+x')(x+y)     by the distributive property
     = 1.(x+y)         by the negation law
     = (x+y).1         by the commutative property
     = x+y             by the identity law

Similar proofs can be made for the other three forms.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/