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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