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

Give the introduction about graphing, Give the  introduction about Graphin...

Give the  introduction about Graphing? Somebody tells you that x = 5 and y = 3. "What does it all mean?!" you shout. Well here's a picture: This picture is what's call

Area under curve, w/ You could use this sample code to test your C function...

w/ You could use this sample code to test your C functions // Please make appropriate changes to use this for C++. // Following main function contains 3 representative test cases

What is fibonacci sequence, what is Fibonacci Sequence? The most famous...

what is Fibonacci Sequence? The most famous sequence in mathematical history is called the Fibonacci sequence, discovered by the 12th-century mathematician Leonardo Fibonacci o

Linear equations, A police academy is training 14 new recruits. Some are wo...

A police academy is training 14 new recruits. Some are working dogs and others are police officers. There are 38 legs in all. How many of each type of recruits are there?

Related to MCA, AskIf y=e^(a?sin?^(-1) x), prove that (1 – x2)yn+2 – (2n + ...

AskIf y=e^(a?sin?^(-1) x), prove that (1 – x2)yn+2 – (2n + 1)xyn+1 – (n2 + a2)yn = 0. Hence find the value of yn when x = 0. question #Minimum 100 words accepted#

Example of distributive law, Maya gives the children examples of distributi...

Maya gives the children examples of distributive with small numbers initially, and leads them towards discovering the law. The usual way she does this is to give the children probl

Pi, is that rational or irrational number

is that rational or irrational number

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