### Incorrect Answer.

Given the predicate logic model

A = {a,b,c,d};
PM = {a,b}
QM = {(a,b), (b,b), (c,b)}
fM (a) = b, fM (b) = b, fM (c) = a, and fM (d) = c

the formula x y (Q(x,y) Q(y,x)) is satisfied in this model if, and only if, for all values of x we can find a value for y such that the pair (x,y) or the pair (y,x) is in QM. Thus, we need to verify this for all possible choices of x.
• If the value of x is a, then (a,b) is in QM and we therefore may choose y to be b.
• If the value of x is b, then (a,b), (b,b), and (c,b) are in QM; therefore may choose y to be b, b, or c, respectively.
• If the value of x is c, then (c,b) is in QM; therefore may choose y to be b.
• Finally, if the value of x is d, there is no pair in QM whose first or second component equals c; therefore we cannot find a value for y to make the disjunction Q(x,y) Q(y,x) true for the given value of x.
Thus, the last case demonstrated that, given that x has value d, the disjunction Q(x,y) Q(y,x) is false for all values of y. Therefore, the formula in question is not satisfied in the given model.
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