--%>
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 : State Fermat algorithm The basic Fermat

    The basic Fermat algorithm is as follows: Assume that n is an odd positive integer. Set c = [√n] (`ceiling of √n '). Then we consider in turn the numbers c2 - n; (c+1)2 - n; (c+2)2 - n..... until a perfect square is found. If th

    		•
  • Q : Problem on augmented matrix Consider

    Consider the following system of linear equations.  (a) Write out t

    		•
  • Q : Bolzano-Weierstrass property The

    The Bolzano-Weierstrass property does not hold in C[0, ¶] for the infinite set A ={sinnx:n<N} : A is infinite; Show that has no “ limit points”.

    		•
  • 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 : 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 : Ordinary Differential Equation or ODE

    What is an Ordinary Differential Equation (ODE)?

    		•
  • Q : What is Big-O hierarchy The big-O

    The big-O hierarchy: A few basic facts about the big-O behaviour of some familiar functions are very important. Let p(n) be a polynomial in n (of any degree). Then logbn is O(p(n)) and p(n) is O(an<

    		•
  • 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 : Set Theory & Model of a Boolean Algebra

    II. Prove that Set Theory is a Model of a Boolean Algebra The three Boolean operations of Set Theory are the three set operations of union (U), intersection (upside down U), and complement ~.  Addition is set

    		•
  • Q : Calculus I need it within 4 hours. Due

    I need it within 4 hours. Due time March 15, 2014. 3PM Pacific Time. (Los Angeles, CA)

    		•