1) $(\exists x \in R)[(x^2 = (x+1)^2 ∧ (x^3 \in Z))]$
ATTEMPT : $((∀ x \in R)[(x^2 \not= (x+1)^2 ∧ (x^3 \notin Z))])$
2)$(∀x \in R)(x>0) ⇒ (\exists n \in N)(n . x >1)$ Note: the (n.x) is multiplication.
ATTEMPT: $(\exists x \in R)(x>0)∧(∀n \in N)(n . x <=1)$
Would be grateful if anyone could look my answers over.
For the first, you need DeMorgan's to get $$(∀ x \in R)[(x^2 \not= (x+1)^2 \lor (x^3 \notin Z))]$$
For the second, we have an implication of the form $p \rightarrow q\equiv \lnot p \lor q$.
To negate it gives us, again, with DeMorgan's, $\lnot(\lnot p \lor q) \equiv p \land \lnot q$. So your quantifier in p should be $\forall$:
$$(\forall x \in R)\Big((x>0)∧(∀n \in N)(nx \leq 1)\Big)$$