Predicate and quantifiers_discrete

Question

Consider the predicate ∀x A(x) → ∃x B(x)

i Just want to know is this predicate always true ? Reason i think "∀x A(x)" will always be false and hence making the predicate always TRUE ..!! let A(x)=x is even B(x)=x is odd now there won't be case when All x will be "even" making it false..please correct me

Answer 1

No, the predicate can be false. Consider $A(x)$ as $x=x$ and $B(x)$ as $x\neq x$. Then your predicate is false.

Answer 2

Reason i think "∀x A(x)" will always be false and hence making the predicate always TRUE ..!!

$\forall x A(x)$ will not always be false, even if maybe it will usually be false. For it to be true, you have to pick $A(x)$ to be something really stupid that is always true, such as "$x = x$" or simply $\textsf{True}$.

let A(x)=x is even B(x)=x is odd now there won't be case when All x will be "even" making it false..

You are right for the specific example of $A(x)$ being "$x$ is even" and $B(x)$ being "$x$ is odd", that the formula comes out true. But for other predicates $A$, $B$, it can come out true. So you haven't shown that it is always true. In fact it is sometimes false, as has already been pointed out.