Logic Web Tutor: Wrong Answer

Incorrect Answer.

The formula set {

x P(x,y),

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

y x P(x,y) and
x P(x,x).

For the first formula to be satisfied in M, we need a value b for y such that (a,b) is in P^M 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 P^M. 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 P^M.

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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