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?

Consider $R={1,2}$ and $A(1),B(2)$ are true while $A(2),B(1)$ are false. What are the truth values of your two expressions? — Prasun Biswas, Oct 21 '20 at 00:54
@PrasunBiswas I was in the middle of writing the same exact thing (lol) — pancini, Oct 21 '20 at 00:57

score 3 · Accepted Answer · answered Oct 21 '20 at 00:57

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.

score 1 · Answer 2 · answered Oct 21 '20 at 00:56

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

Difference between usages of ∀

2 Answers2