Question 384304
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1.   x -> (x * y)
2.   ((x * y) v (~x * y)) -> y           / y -> y
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3.   (y * (x v ~x)) -> y                      2   Distribution
4.   (y * T) -> y                             3   See note1 below
5.   y -> y                                   4   See note2 below
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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: 


<pre>
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
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