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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)

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