Absorption LawsDate: 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/ |
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