Is there a difference between these two statements? one uses parenthesis, while the other doesn't.
$(\exists x\in \mathbb{R}, x>5) \Rightarrow (\forall y \in \mathbb{R}, y>-100)$
is it the same as
$\exists x\in \mathbb{R}, x>5 \Rightarrow \forall y \in \mathbb{R}, y>-100$
- Are these two equivalent?
- are both false-statements?
It is a matter of convention whether the missing parentheses in $\exists x, P \Rightarrow \forall y, Q$ should be inserted to give $(\exists x, P) \Rightarrow (\forall y, Q)$ or to give $\exists x, (P \Rightarrow (\forall y, Q))$. The former reading is most common in the mathematical logic literature, but the latter reading agrees with the usual conventions for the $\lambda$-calculus and has been adopted in many mechanized theorem-proving systems such as HOL or Coq.
In the case in point, the two readings are not equivalent:
In $(\exists x\in \mathbb{R}, x>5) \Rightarrow (\forall y \in \mathbb{R}, y>-100)$, the left-hand side of the implication is true (e.g., take $x = 6)$ while the right-hand side is false (e.g., take $y = -100$). So the overall predicate is false.
However, in $\exists x\in \mathbb{R}, (x>5 \Rightarrow (\forall y \in \mathbb{R}, y>-100))$, we can take $x = 5$, to make $x > 5$ false and hence the overall predicate is true, since a false statement implies any other statement and we can find an $x$ that falsifies $x > 5$.