Difference between usages of ∀

Question

Is (∀n ∈ R, A(n)) ∨ (∀n ∈ R, B(n)) the same as ∀n ∈ R (A(n)) ∨ B(n)). My belief is that it should be the same no?

Answer 1

No, they are not the same. Suppose that $A(n)$ is ‘$n$ is even’, $B(n)$ is ‘$n$ is odd’, and $R=\Bbb N$. Then

$$\forall n\in R\,\big(A(n)\lor B(n)\big)$$

says that every natural number is even or odd, which is true, while

$$\left(\forall n\in R\,\big(A(n)\big)\right)\lor\left(\forall n\in R\,\big(B(n)\big)\right)$$

says that either every natural number is even, or every natural number is odd, which is false.

Answer 2

These are not the same. Suppose $R=\{1,2\}$, for example, and that \begin{align} A(1)&=\text{true}\\ A(2)&=\text{false}\\ B(1)&=\text{false}\\ B(2)&=\text{true} \end{align}

Then for any $n\in R$, it's true that either $A(n)$ or $R(n)$ it true. However, it is is not true that either

• $A(1)$ and $A(2)$ is true, or
• $B(1)$ and $B(2)$ is true