Looking at a computer-science book, I see that a + -a = 0 (inverse property in integer ring), and that a & ~a = 0 (Boolean complement).
But these are not given as equivalent (or shared properties. But on the face of it, they seem to be.
Why are they not equivalent?
In set theory, you do not have this.
$$ A - B = A - C \implies B = C;$$ This merely tells you $A\cap B = A\cap C$. Outside of $A$ it tells you nothing. The same thing happens in logic.