Incorrect Answer.


The formula set { y x P(x,y), x P(x,x)}. 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 two formulas For the first formula to be satisfied in M, we need a value b for y such that (a,b) is in PM for all values a. In particular, this has to be the case for the value of y. Thus, we infer that (b,b) is in PM. But 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 two formulas. (Can you find models that satisfy just one of these formulas?)
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