--%>
Assignment Help - homework help

Abstract Boolean Algebra

I. Boolean Algebra

Define an abstract Boolean Algebra, B,  as follows:

 The three operations are:

 +   ( x + y addition)

  • ( x y multiplication)~

˜ ( ˜ x  the complement  or the negation of x)

{B, + , 0 } is a commutative monoid

1. State the commutative law of addition: ___________________________________________

2. State the associative law of addition: _____________________________________________

3. State the law that says 0 is an additive identity __________________________________

{B, • , 1 } is a commutative monoid

4. State the commutative law of multiplication: ____________________________________

5. State the associative law of multiplication: _______________________________________

6. State the law that says 1 is a multiplicative identity _____________________________

7. State the distributive law of multiplication: ______________________________________

8. State the distributive law of addition: _____________________________________________

Finally  it is given that:

9.   x  +  ˜ x  = 1

10. x  •  ˜ x  = 0

The above ten properties are necessary and sufficient conditions to prove a given algebra is a Boolean algebra.

For a Boolean Algebra prove the idempotent properties:

1.  x  •  x  = x 

2.  x  +  x  = x 

For a Boolean Algebra prove the Zero and One Properties:

3.  0  •  x  = 0 

4.  1  +  x  = 1  

Prove the four Absorption Laws for a Boolean Algebra:

5.  x + (x  • y) = x 

6.  x  • ( x +  y) = x  

7.  x  +  (˜x • y) = x + y 

8.  x  • ( ˜x +  y) = x  •  y 

9. Prove that if the element y acts as the additive complement of x, i.e. x + y = 1, and y acts as the multiplicative complement of x, i.e. x•y = 0, then in fact x is the complement of y, i.e.  y =  ˜x.

Note.  The Involution Law:  ˜ ˜x = x, is true, by the fact of the uniqueness of the complement (see 9. above) and the fact that x acts as the complement of ˜x . 

Prove the following De Morgan Laws (Hint:  use the uniqueness of the complement)

10.  ˜ ( x + y ) = (˜x)  • (˜y)

11.  ˜ ( x + y ) = (˜x)  • (˜y)

   Related Questions in Mathematics

  • Q : Formal logic2 It's a problem set, they

    It's a problem set, they are attached. it's related to Sider's book which is "Logic to philosophy" I attached the book too. I need it on feb22 but feb23 still work

    		•
  • Q : Test Please read the assignment

    Please read the assignment carefully and confirm only if you are 100% sure. Please go through below mentioned guidelines and penalties: • Your solution must be accurate and complete. • Please do not change Subject Title of the Email. • Penalty clause will be applied in case of delayed or plag

    		•
  • Q : Numerical solution of PDE this

    this assignment contains two parts theoretical and coding the code has to be a new. old code and modified code will appear in the university website .

    		•
  • Q : Problem on mass balance law Using the

    Using the mass balance law approach, write down a set of word equations to model the transport of lead concentration. A) Draw a compartmental model to represent  the diffusion of lead through the lungs and the bloodstream.

    		•
  • Q : Competitive equilibrium 8. Halloween is

    8. Halloween is an old American tradition. Kids go out dressed in costume and neighbors give them candy when they come to the door. Spike and Cinderella are brother and sister. After a long night collecting candy, they sit down as examine what they have. Spike fi

    		•
  • Q : Formal logic It's a problem set, they

    It's a problem set, they are attached. it's related to Sider's book which is "Logic to philosophy" I attached the book too. I need it on feb22 but feb23 still work

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • Q : Solve each equation by factoring A

    A college student invested part of a $25,000 inheritance at 7% interest and the rest at 6%.  If his annual interest is $1,670 how much did he invest at 6%?  If I told you the answer is $8,000, in your own words, using complete sentences, explain how you

    		•
  • Q : State Prime number theorem Prime number

    Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often diffic

    		•
  • Q : Probability and Stochastic assignment

    Introduction to Probability and Stochastic Assignment 1: 1. Consider an experiment in which one of three boxes containing microchips is chosen at random and a microchip is randomly selected from the box.

    		•