Question 384304: I need a Formal Logic Proof for the following (I know that it is valid):
(P1) X⊃(X•Y)
(P2) [(X•Y)∨(~X•Y)]⊃Y / Y⊃Y
*I am aware that there are many different variations on the symbols, etc. used in formal logic, so below are the symbols we use in class:
~ negation
∨ or
• and
≡ if and only if
⊃ if ... then
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
1. x -> (x * y)
2. ((x * y) v (~x * y)) -> y / y -> y
-----------------------------------------------
3. (y * (x v ~x)) -> y 2 Distribution
4. (y * T) -> y 3 See note1 below
5. y -> y 4 See note2 below
Note1: Recall that p v q is only true if either p or q (or both) are true. Now say that we want to simplify x v ~x. If x is true, then x v ~x is automatically true. On the other hand, if x is false, then ~x is true (since its the opposite of false). So x v ~x is still true. So x v ~x is ALWAYS true regardless of the truth value of x. We usually denote this with a capital T.
Note2: Since p * q is only true if BOTH p and q are true, this means that y * T is wholly dependent on the value of y since T is a constant. Basically, this is saying that y * T = y
Alternative Answer:
1. x -> (x * y)
2. ((x * y) v (~x * y)) -> y / y -> y
-----------------------------------------------
3. ~x v (x * y) 1 Material Implication
4. (~x v x) * (~x v y) 3 Distribution
5. ~x v x 4 Simplification
6. (y * (x v ~x)) -> y 2 Distribution
7. ((x v ~x)*y) -> y 6 Commutation
8. (x v ~x) -> (y -> y) 7 Exportation
9. (~x v x) -> (y -> y) 8 Commutation
10. y -> y 9,5 Modus Ponens
|
|
|
