--%>
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 : Elasticity of Demand For the demand

    For the demand function D(p)=410-0.2p(^2), find the maximum revenue.

    		•
  • Q : Econ For every value of real GDP,

    For every value of real GDP, actual investment equals

    		•
  • Q : Area Functions & Theorem Area Functions

    Area Functions 1. (a) Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t - axis, and between the vertical lines t = 1 and t = 3. (b) If x > 1, let A(x) be the area of the region that lies under the line y = 2t + 1 between t

    		•
  • Q : Explain Factorisation by trial division

    Factorisation by trial division: The essential idea of factorisation by trial division is straightforward. Let n be a positive integer. We know that n is either prime or has a prime divisor less than or equal to √n. Therefore, if we divide n in

    		•
  • 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 : Define Well-formed formulas or Wffs

    Wffs (Well-formed formulas): These are defined inductively by the following clauses:    (i) If  P  is an n-ary predicate and  t1, …, tn are terms, then P(t1, …, t

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • Q : Elementary Logic Set & Model of a

    Prove that Elementary Logic Set is a Model of a Boolean Algebra The three Boolean operations of Logic are the three logical operations of  OR ( V ), AN

    		•
  • Q : First-order formulas over the

    Consider the unary relational symbols P and L, and the binary relational symbol On, where P(a) and I(a) encode that a is apoint and a (sraight) line in the 2-dimensional space, respectively, while On(a,b) encodes  that a is a point, b is a line, and o lies on b.

    		•
  • Q : Probability assignments 1. Smith keeps

    1. Smith keeps track of poor work. Often on afternoon it is 5%. If he checks 300 of 7500 instruments what is probability he will find less than 20substandard? 2. Realtors estimate that 23% of homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2

    		•