If $x$ and $y$ are integer variables that have value $0$ or $1$, then what does the expression $x + y - xy$ mean? Answer Options:
$1.$ Logical AND
$2.$ Logical OR
$3.$ Nothing, it makes no sense
$4.$ Logical implies
I'm stuck between Logical OR and "Nothing, it makes no sense" because when I create a truth table for $x  + y - xy$, it is equivalent to Logical OR.
But also, "$-$" doesn't really mean anything in boolean logic, right?
Please help me out here.
 A: You're right that "$-$" doesn't really have an interpretation as a binary operation of booleans, but, as the truth table shows, this whole expression does make sense even if it contains a component that does not. Subtraction does make sense in the unary operation that $1-x$ is "not $x$" in booleans.
Really, what is going on here is that we have the boolean identity:
$$x\text{ or }y=\text{not }((\text{not } x)\text{ and }(\text{not }y))$$
and the right hand side encodes as
$$1-(1-x)(1-y)$$
where all the subtractions have a clear interpretation. This just happens, in the world in integers, to be the same as $x+y-xy$ after distributing - even though this doesn't really make sense in the world of booleans.
A: The answer is OR.
Boolean equations can be represented as polynomials.
Each boolean variable satisfies the equation $x(1-x) = 0$. The only solutions are $x = 0 \ or \:1$.
$\text{not} \: x = 1 - x$
$x \: \text{and} \: y = xy$
$x \: \text{or} \: y = x + y - xy$
$x \: \text{xor} \: y = x + y - 2xy$
A: Considering the operations $+$ and $-$ from the usual field of real numbers, you have the following output for $x+y-xy$.
(a) If $x=y=0 $  then output is $0$.
(b) If $x=1,y=0 $  then output is $1$.
(c)If $x=0,y=1$  then output is $1$.
(d)If $x=y=1 $  then output is $1$.
This resembles the truth table of logic 'OR'.
A: Given those four options I'd say logical OR
But if I could say what I'd think I'd say none of those.  $x + y -xy$ makes perfect sense; it means "add $x$ and $y$ and subtract $x$ times $y$".  It means what it means nothing more nothing less.
It can be interpreted and is equivalent to the logical OR but it doesn't mean the logical OR

But also, "−" doesn't really mean anything in boolean logic, right?

but note, the question never said that $x + y - xy$ was an expression of boolean logic.  In fact, in describing $x,y$ as "integer variables that have value 0 or 1"and not as "variables representing boolean bits" it is to be understood that this is a real operation.
But if the domain is restricted to $\{0,1\}$ the operation is designed to restrict the range to $\{0,1\}$ too.
(It's kind of clever the way it does that; if either $x,y$ are $0$ then $xy=0$ and has no significance.  And as $0$ as an additive identity the value is the same as any non-zero variable if any.  If both $x$ and $y$ are $1$ then $x+y=2$ which "overflows" but $xy=1$ so that "tamps it back down" so $xy$ acts as a pressure release....)
And interpreted as a BOOLEAN function it's value directly correspond with the logical OR.
(And maybe I'm rambling but I rather like how this can be interpreted as "or". If both of $x,y$ are $0$ then nothing "triggers" it.  If one is $0$ and the other is $1$ then the $1$ value is "passed through".  It's cludgy if $x,y$ are both $1$ but the pressure release sets it back to one canceling either the $x$ or the $y$ but letting something equal to $y$ or $x$ through.  So the "truthier value passes through.  ... Or equivalently it is only triggered if one or the other or both is true.  And that is what naturally "or" means.  .... Okay, that was rambling and handwaving but... I'm a weirdo and I'm easily amused.)
