TITLE: All prime numbers are either even or odd, Is it a true statement?
QUESTION [7 upvotes]: $r:$All prime numbers are either even or odd, Is it a true statement?


I was studying Mathematical Logic then i came across above question.
Since here connecting word is "OR"
so if i separate two statement then it become
$p:$ All the prime number are even
$q:$ All the prime number are odd
Because both statement $p$ and $q$ are false so final statement $r$ must be false using truth value of statement for "OR" connective.
But my intuition says $r$ is true.
Am i thinking correct?
Please Help me in this.

REPLY [2 votes]: $✔\quad$ All numbers are either even or odd.
$✗\quad$ Either all numbers are even, or all numbers are odd.
In general, $$∀x \;\Big(A(x)\text{ or }B(x)\Big)\quad\text{does not imply}\quad∀x A(x)\;\text{ or }\;∀x B(x);$$ however, $$∃x \;\Big(A(x)\text{ or }B(x)\Big)\quad\text{is equivalent to}\quad∃x A(x)\;\text{ or }\;∃x B(x),$$ and $$∀x \;\Big(A(x)\text{ and }B(x)\Big)\quad\text{is equivalent to}\quad∀x A(x)\;\text{ and }\;∀x B(x).$$