Elementary Logic as a Model of a Boolean Algebra Elementary logic consists of al

Question

Elementary Logic as a Model of a Boolean Algebra Elementary logic consists of all logical statements which can be considered to have a value either true, T, or false, F, for example the statement P: "I like cherry pie." Can be considered either true or false depending on who is speaking. The same goes for the statement Q: "I like vanilla ice cream. "On the other hand. The triangle is green is not considered to be either true or false; actually it makes no sense. The sentences can be connected by two binary operators, OR symbolized by v, AND symbolized by A, and the unary operator NEGATION symbolized by~. Depending on the values of the statements joined by operators, the composite statement has a value true, T, or false F, which is stated in a truth table. For purposes of the next question use only the logical operators v, A, and ~. Use T for true and F for false. Just state the Boolean algebra equalities of logical statements below, the proof are considered self-evident and/or tedious. Let P, Q, R denote arbitrary statements instead of, y, z, etc. And use the logical operators listed above. State the commutative law of addition: State the associative law of addition: State the law that says empty set is an additive identity State the commutative law of multiplication: State the associative law of multiplication: State the law that says S is a multiplicative identity State the distributive law of multiplication: State the distributive law of addition: State the Boolean algebra property x + (-^x) = 1 in terms of a set A. State the Boolean algebra property x middot (-^x) = 0 in terms of a set A. These statements prove Set Theory is a model of a Boolean algebra. Rewrite the logical statement ~(P Q) in terms of the negation of P and the negation of Q individually. ~(P Q) =

Explanation / Answer

1)
P + Q = Q + P (or)
P v Q = Q v P

2)
P + (Q + R) = (P + Q) + R (or)
P v (Q v R) = (P v Q) v R

3)
P + 0 = P
P v Phi = P

4)
P * Q = Q * P

5)
(P * Q) * R = P * (Q * R)

6)
P * 1 = P

7)
P(Q + R) = PQ + PR

8)
P(Q + R) = PQ + PR

9)
It's a OR of P and P's Compliment

P v P' = 1

10)
It's a AND of P and P's Compliment

P cap P' = 0

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