Define domain $X,$ predicate $A(X)$ and $B(X)$

Question

I'm having trouble creating a domain $X$ and the predicates $A(X)$ and $B(X)$ to for this set of sentences to be evaluated to be true or false.

$(T)\quad \forall x \in X, (A(x) \rightarrow B(x))$

$(F)\quad \exists x \in X (A(x) \land B(x))$

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I've tried letting X be the set of all natural numbers, A(x): "x is even", B(x): "x is odd" as it worked in past questions. However, I'm stumped on this one. I also want to know if I can use a more generic domain and predicate rather than being limited to numbers? If so, how? Any help is appreciated!

Answer 1

Define $A(x):$ "x is $\dfrac 13$".

Define $B(x):$ "x is an integer."

And let the domain be the set of integers. Then the first sentence is true, but the second sentence is false. The first sentence is true vacuously, because $\frac 13 \notin \mathbb Z$, and an implication is true whenever the antecedent is false. The second is false, since $\lnot \exists x \in \mathbb Z$ such that $x = \frac 13$, and a conjunction is true if and only if both conjuncts are true.