Probably Intuition never fails, We fail Intuition

When I say, ¨ $\phi$ is the least upper limit to the numerical error in taking $p$ to represent $P$ ¨, where $p$ and $P$ are real numbers, let me mean—

$\phi$ is the minimum positive real number so that $p-\phi \leq P \leq p+\phi$ .

Now assume that

• $\alpha$ is the least upper limit to the numerical error in taking $a$ to represent $A$ .
• $\beta$ is the least upper limit to the numerical error in taking $b$ to represent $B$ .

You can verify, under those assumptions, that

$\alpha + \beta$ is the least upper limit to the numerical error in taking $a+b$ to represent $A+B$ .

Is the following assertion, under the same two assumptions, TRUE or FALSE ?

$\alpha - \beta$ is the least upper limit to the numerical error in taking $a-b$ to represent $A-B$ .