I have found two different statements on axiom of foundation of Zermelo–Fraenkel set theory in two different books as:
1) every nonempty set contains an element that is not an element of any other element in the set.
2) Every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets.
These two are not equivalent statements. Am I correct ?
They are not equivalent. In fact, (1) is actually refutable under a weak subtheory of ZF. Consider any inductive set $X$. Then for each $u \in X$ we have that $u \in u \cup \{ u \} \in X$. Thus $X$ has no element as described in (1).