--%>
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 : Problem on reduced row-echelon The

    The augmented matrix from a system of linear equations has the following reduced row-echelon form. 280_row echelon method.jpg

    		•
  • Q : Where would we be without stochastic

    Where would we be without stochastic or Ito^ calculus?

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • 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 : Logic and math The homework is attached

    The homework is attached in the first two files, it's is related to Sider's book, which is "Logic for philosophy" I attached this book too, it's the third file.

    		•
  • 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 : Theorem-G satis es the right and left

    Let G be a group. (i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b. (ii) If g ≡ G, then (g-1)

    		•
  • Q : Problem on Nash equilibrium In a

    In a project, employee and boss are working altogether. The employee can be sincere or insincere, and the Boss can either reward or penalize. The employee gets no benefit for being sincere but gets utility for being insincere (30), for getting rewarded (10) and for be

    		•
  • Q : Mathematical Method for Engineers The

     The function is clearly undefined at , but despite all of this the function does have a limit as approaches 0. a) Use MATLAB and ezplot to sketch for , and use the zoom on facility to guess the . You need to include you M-file, outp

    		•
  • Q : Maths assignment complete assignment

    complete assignment with clear solution and explanation

    		•