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

Area and surface Are, I cant figure out how to study for my math test

I cant figure out how to study for my math test

Population problem - nonhomogeneous systems, The next kind of problem seems...

The next kind of problem seems as the population problem. Back in the first order modeling section we looked at several population problems. In such problems we noticed a single po

Velocity of a particle, A particle moves along a straight line so that afte...

A particle moves along a straight line so that after t secs its distance from fixed point O on the line is given by s=(t-1)^2(t-2).find the distance from O when the velocity is zer

Ratio, There are only Chinese and Malay pupils in a hall.The ratio of the n...

There are only Chinese and Malay pupils in a hall.The ratio of the number of boys to the number of girls is 2:3.The ratio of the number of Chinese boys to the number of Malay boys

Difference between experiment and outcome, Difference Between Experiment an...

Difference Between Experiment and Outcome Experiment is an operation that produces outcomes which can be observed. Outcome/Event is the result of an experiment.

Fractions, Mr. And Mrs. samuel visited Florida and purchased 120 oranges. ...

Mr. And Mrs. samuel visited Florida and purchased 120 oranges. They gave 1/4 of them to relatives, ate 1/12 of them in the hotel, and gave 1/3 of them to friends. The shipped the

Customary units of length, Eileen needs 9 feet of fabric to make a skirt. I...

Eileen needs 9 feet of fabric to make a skirt. If Eileen has 18 feet of fabric how many skirts can she make?

Harmonic progression (h.p.), Three quantities a, b and c are said to ...

Three quantities a, b and c are said to be in harmonic progression if, In this case we observe that we have to consider three terms in o

The rank correlation coefficient (r), The Rank Correlation Coefficient (R) ...

The Rank Correlation Coefficient (R) Also identified as the spearman rank correlation coefficient, its reasons is to establish whether there is any form of association among tw

Indeterminate form, Indeterminate form : The 0/0 we initially got is calle...

Indeterminate form : The 0/0 we initially got is called an indeterminate form. It means that we don't actually know what it will be till we do some more work.  In the denominator

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