Can this be true? $\exists x P(x) \wedge \exists x \neg P(x) $

Question

Can the following be true or is it a contradiction?

$ \exists x P(x) \wedge \exists x \neg P(x) $

Sorry if this is an obvious question, i'm new to this topic.

Answer 1

The important point is that the two $x$s are different. They are each dummy variables that extend from the $\exists$ to the $P(x)$. A far better way to write the sentence is $$\exists x P(x) \wedge \exists y \neg P(y)$$ This is logically equivalent to your expression, but it avoids the confusion caused by reusing variables. In this version, it is easy to see that the sentence can be true,

Answer 2

Of course this can be true. Example:-

P(x) : x is prime

Answer 3

Consider $P(x)$ be the statement that "$x \in\mathbb R$ and $x=1$".