Is plus sign the correct notation for Disjunction in boolean algebra? As far as I know, the disjunction is notated as $\vee$ in boolean algebra
However, in some context, I saw that people usually tend to use the plus sign + to notate the disjunction like this:
$$f(x, y) = x + y + xy $$
Instead of:
$$f(x, y) = x \vee y \vee xy $$
Since the disjunction in boolean algebra is explicitly defined: $x \vee y = x + y - xy $, in order to express this logical formula with the ordinary operations of arithmetic, I don't think the $\vee$ sign (logical operator) can be replaced by the plus sign (arithmetical operator) is mathematically correct.
 A: You are right that if we take the $+$ as an arithmetical operator, then we should capture $x \vee y$ with the arithmetic expression $x + y - xy $.  Of course, to say that $x \vee y= x + y - xy$ we should then probably see the $\lor$ as an arithmetical operator as well ... though it is now not clear what the value of something like $2 \lor 2$ is ...
However, when we use the '$+$' in the context of boolean algebra, then its operation has been redefined, so that we no longer have that $1+1=2$, but  $1+1=1$. Indeed, in boolean algebras we only have $0$ and $1$ as the only possible values of any expression., so it wouldn't even make sense to have $1+1=2$.
In other words, we repurposed the $+$ sign, and the way it works in boolean algebra is not the same as it works in arithmetic.
So for that repurposed operator, we really have that $x \lor y$ is the same as $x + y$.  But again, yes, if the $+$ is understood as the original arithmetical operator working in the domain of numbers, then we would have to use $x + y - xy $ to capture the $\lor$ ... as constrained for $0$ and $1$'s at least.
