Prove complement of element in boolean algebra is unique, Mathematics

Assignment Help:

Prove that, the complement of each element in a Boolean algebra B is unique.    

Ans:  Proof: Let I and 0 are the unit and zero elements of B correspondingly. Suppose b and c be two complements of an element a ∈ B. After that from the definition, we have 

a ∧ b = 0 = a ∧ c and

a ∨ b = I = a ∨ c 

We can write b = b ∨ 0 = b ∨ (a ∧ c )

= (b ∨ a) ∧ (b ∨ c)   [as lattice is distributive ]

= I ∧ (b ∨ c )

=  (b ∨ c )

Likewise, c = c ∨ 0 = c ∨ (a ∧ b )

= (c ∨ a) ∧ (c ∨ b)   [as lattice is distributive]

= I ∧ (b ∨ c)   [as ∨ is a commutative operation]

=  (b ∨ c)

The above two results define that b = c.


Related Discussions:- Prove complement of element in boolean algebra is unique

Finding the equation of a line, Finding the Equation of a line, Given a Slo...

Finding the Equation of a line, Given a Slope and a Point ? Find the equation of a line with slope m = 2, which passes through the point (-1, -3). Solution: Use the po

One of these food groups, In a collection of 30 dissimilar birds, 15 eat wo...

In a collection of 30 dissimilar birds, 15 eat worms, 18 eat fruit, and 12 eat seeds. Accurately 8 eat worms and seeds, 8 eat worms and fruit, 7 eat fruit and seeds, and 4 eat each

Determine probability , You are going on a road trip and you buy snack pack...

You are going on a road trip and you buy snack packs and three different kind of beverages.  You buy 7 Cokes, 5 Pepsis and 4 Dr. Peppers.  You pull out two beverages at random.  An

Systems of linear equation, a man can row a bangka at a rate of 5 km/h in s...

a man can row a bangka at a rate of 5 km/h in still water. It takes 10 minutes longer to row upstream a distance of 2km than he takes to row downstream. What is the rate of the cur

Decimals, 0.875 of a number is 2282. What is the number ?

0.875 of a number is 2282. What is the number ?

Give the introduction to amino acid and nucleotide metabolis, Give the Intr...

Give the Introduction to amino ACID and nucleotide metabolism ? Here, we studied about the chemistry of proteins and amino acids. We studied that the amino acids are used for p

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd