Why does $AB + BC + CA = (A\oplus B)C + AB$ not imply $BC + CA = (A\oplus B)C$ in boolean algebra? I am new to Logical Inequalities. Please bear with me if I am inexplicably stupid.
The following is a Proven Equality:
$$AB + BC + CA = (A\oplus B)C + AB$$
I noticed that I cannot "cancel out" $AB$ from both sides of the identity, i.e.
$$BC + CA \neq (A\oplus B)C$$
In mathematical equations, we cancel out stuff all the time. For example,
$$X + 3 = Y + 3 \quad\implies\quad X = Y$$
Can this not be done with Logical Statements? Is there a reason why this is such a way?

It is worth clarifying here: there is only one single rule at all times for equations: If $x=y$, then $f(x) = f(y)$, where $f$ is some mapping. Nothing more, nothing less.
What about cancellation? Isn't it true that, if $f(x) = x+ 3$, then $f(x)=f(y)$ implies $x = y$? Does this make a rule?
No. Forget it. What really happens behind this is that we define $g(x)=x-3$, so
$g(f(x)) = x$. And we apply the one single rule and we get $g(f(x)) = g(f(y))$, which simplifies to $x=y$. Is that clear?
Now go ahead and find a function $g$ such that when composed with $f(x) = x \oplus A$ recovers $g(f(x))=x$. There is no such function. And this is why cancellation doesn't hold.
