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Elementary Logic Set & Model of a Boolean Algebra

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 ), AND (upside down V), and NEGATION ~.  Addition is the logical OR , multiplication is the logical AND, and complement is the logical NEGATION.  The symbol 1 is the logical T (True), and the symbol 0 is the logical F (False) . (Just state the Boolean Algebra versions of logical statements below, the proofs are considered self-evident, we do not require Truth Tables to be written to establish their validity.)

1. State the commutative law of addition: _________________________________________

2. State the associative law of addition: ___________________________________________

3. State the law that says F is an additive identity __________________________________

4. State the commutative law of multiplication: _____________________________________

5. State the associative law of multiplication: _______________________________________

6. State the law that says T is a multiplicative identity _______________________________

7. State the distributive law of multiplication: _______________________________________

8. State the distributive law of addition: ____________________________________________

9.   State the Boolean Algebra property x  +  ˜ x  = 1 in terms of a logical statement A.

 10.   State the Boolean Algebra property x  •  ˜ x  = 0 in terms of a logical statement A.

The above ten properties are necessary and sufficient conditions to prove that Elementary Logic is indeed a model of a Boolean algebra.

11. In Elementary Logic, A implies B ( A-> B), has a Truth table, which we recall is only False (F), when B is False and A is True.  Rewrite the logical statement

A -> B in terms of the basic logical operations of AND (upside down V, we will have to use in this document the symbol ?), OR (V) and NEGATION (~).

A -> B =   

12. In terms of an Abstract Boolean Algebra, for two elements x and y define that x implies y,  x -> y using the basic operations  +,  •, and ~ of  Boolean Algebra, using the definition from Elementary Logic as your guide.

x -> y  

Recall that in Elementary Logic a Tautology is a statement which is always True, regardless of the truth values of its constituent statements., e.g.  A V ~A .

13. Write the Truth table for the logical statement (A->B)  V (B->A).   

Is (A->B)  V (B->A)  a tautology?

14. Write the Truth table for the logical statement  (B ? (A->B) ) ->A  (recall ? is unfortunately our symbol for AND, the upside down V).   

Is (B ? (A->B) ) ->A a tautology?

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