Define an abstract Boolean Algebra B as follows The three op

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 = ________________________________

10. x • ~ x = ________________________________

Solution

{B, + , 0 } is a commutative monoid

1. State the commutative law of addition: B+0=0+B=B

2. State the associative law of addition:(B+0)=(0+B)

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

    if x+0=x   and x+0\'=x so x+0=x+0\'   and therefore, by cancellation, 0=0\'

4. State the commutative law of multiplication:B*0=0*B=B

5. State the associative law of multiplication:(a × b) × c = a × (b × c)

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

     The number 1 is called the multiplicative identity because multiplying any number a by 1 just gives back a :

                    1 · a = a

7. State the distributive law of multiplication:

The Distributive Law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately
    Example: 3 × (2 + 4) = 3×2 + 3×4

8. State the distributive law of addition:

The Distributive Law says that adding a number by a group of multiplication numbers together is the same as doing each addition separately


Example: 3×2 + 3×4=3 × (2 + 4)

9. x + ~ x = 0

10. x • ~ x = ~ x