Incorrect Answer.
The formula set
{ 
x y (P(x,y) 
P(y,y)), x 
P(x,x), 
x y (P(x,y)}
is satisfied in a model M, if all the formulas in that set are satisfied by
that same model. Thus, we seek a model, M, that satisfies the three formulas
- x y (P(x,y) P(y,y))
- x P(x,x) and
- x y (P(x,y).
For the third formula to be satisfied in M, we need values, a for x and
b for y, such that (a,b) is in PM. But then we can infer
that (a,a) is in PM as well, provided that our model satisfies the
first formula. However, the latter
makes it impossible
for M to satisfy the second formula, which states (if re-expressed with
quantifier equivalences) that no pair of the form (a,a) is in PM.
Thus, there cannot be a model M that satisfies all three formulas. (Choosing
any two formulas above, can you find models that satisfy these two formulas?)
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