∃xA(x) ∧ ∃x¬A(x) is this formula satisfiable ?

Question

I am little confused regarding satisfiabilty problems. can someone help me to understand whether ∃xA(x) ∧ ∃x¬A(x) is satisfiable or not ? and if yes what is the minimum number of elements in each domain model ?

Cheers

Answer 1

You can take the set containing two elements, $\{x,y\}$. And let $A=\{x\}$. Now there is an $x$ satisfying $A(x)$, namely $x$. There also is a x satisfying $\neg A(x)$, namely $y$.

If the sentence would have been $\exists x(A(x)\wedge\neg A(x))$, it would not have been satisfiable.

Answer 2

Hint: Intuitively, this means there is some x making A(x) true, and some x making A(x) false. Is this possible? If it is, can they be the same x?