### Correct 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 (Q(f(x),x) Q(x,x)) is satisfied in this model if, and only if, for all values of x such that the pair (fM(x),x) is in QM we also have that (x,x) is in QM. To determine whether this formula is satisfied in the given model, we only have verify the validity of the implication for all choices of x, where (fM(x),x) is in QM (why?).
• If the value of x is a, then the resulting pair (fM(a),a) is (b,a) and not in QM. There is nothing to check in this case.
• If the value of x is b, then the resulting pair (fM(b),b) is (b,b) which is in QM. Thus, we have to check whether (b,b) is in QM as well, but we already noted that.
• If the value of x is c, then the resulting pair (fM(c),c) is (a,c) is not in QM. There is nothing to check in this case.
• Finally, if the value of x is d, then the resulting pair (fM(d),d) is (c,d) which is not in QM. There is nothing to check in this case.
To summarize, we showed that the implication Q(f(x),x) Q(x,x) holds for all choices of x. Thus, the formula in question is satisfied in the given model.
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