Question

[DISCRETE MATHEMATICS]
Specify a predicate P(x, y) over the set of integers so that all the following conditions are satised

Answer 1

Specify a predicate P(x, y) over the set of integers so that all the following conditions are satised:
$$\forall x\ \exists\ y\ P(x,y) = F_0 $$
$$\exists x\ \forall\ y\ P(x,y) = F_0 $$
$$\forall y\ \exists\ x\ P(x,y) = T_0 $$

Your answer should clearly
specify a predicate P(x; y) and explain why the predicate satises all the three conditions.

Answer 2

I believe a predicate could be
x≠y


For the first condition, since x and y are both over the set of integers, at some point for every x, we will come across
x=y



For the second condition, since x and y are both over the set of integers, there is NO x that exists such that for every y $$ x \not = y $$ because for every x that exists, there is one corresponding y that satisfies $$ x = y $$ and therefore makes $$ (x \not = y) = F_0 $$

For the third condition, since x and y are both over the set of integers, for every y there exists an x such that
x≠y
which would be satisfied for any integer except when
x=y


Is this answer correct and justified?