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?