suppose P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether the following statements are true.
-$$\neg \exists x \exists y ( P ( x , y ) \wedge \neg P ( y , x ) )$$

Question

Quizplus · Accepted Answer

The statement $\neg \exists x \exists y ( P ( x , y ) \wedge \neg P ( y , x ) )$ is false because there exists at least one pair $(x, y)$ such that $P(x, y)$ is true and $P(y, x)$ is false. For example, $P(1, 3)$ is true, but $P(3, 1)$ is not false (since $P(3, 1)$ is actually true in the given conditions, a better counterexample would be a pair where $P(x, y)$ is true and $P(y, x)$ is not listed as true, but based on the given information, all true $P(x, y)$ have their $P(y, x)$ also true, which means the explanation should focus on the misunderstanding of the predicate conditions given). Given the conditions, actually, there isn't a pair where $P(x, y)$ is true and $P(y, x)$ is false, which means the initial explanation was incorrect in its interpretation. The correct interpretation should note that all given true instances of $P(x, y)$ have their counterparts $P(y, x)$ also true or not mentioned as exceptions, thus not providing a clear counterexample to the statement. The essence of the statement is to find if there's a contradiction in the existence of a one-way relationship that isn't reciprocated, based on the given true instances.

Suppose P(x, Y) Is a Predicate and the Universe for the Variables