It came accros my mind when doing an exercise in calculus. Consider the following inequalities:
$1\leq 2\leq 5$
$1\leq 3\leq 5$
Adding the equaions up is totally fine. But subtracting them, we get:
$0\leq 2-3\leq 0$ ??
I fail to understand what is wrong with this step. If someone could formally explain it, it would be greatly appreciated.
On
The reason inequalities are preserved under addition is that a sum of non-negative numbers is non-negative. So from $5-2\geq0$ and $5-3\geq0$ you get $10-5=(5-2)+(5-3)\geq0$, so $5\leq 10$.
But there's no such rule for the difference of non-negative numbers. The fact that $2$ and $3$ lie between $1$ and $5$ doesn't tell you anything about their relationship. This should be obvious from the fact that should you change the order of the inequalities, you get $3-2$ instead of $2-3$.
On
Subtracting is, by definition, adding the negative. a- b= a+ (-b).
But if 1< 3< 5 it is NOT true that -1< -3< -5. Instead, -5< -3< -1.
On
The subtraction is equivalent to multiplying the second line by $-1$, then adding it to the first line. However, the first operation changes the inequalities in the second line to $$ -1\geq -3 \geq -5, \quad\text{or}\ -5 \leq -3 \leq -1. \tag{1} $$ Adding $(1)$ to $1\leq 2\leq 5$, you now obtain the valid result $$ -4\leq -1 \leq 4. \tag{2} $$
When dealing with inequalities, addition of inequalities that are both true is a valid operation that preserves the truth of the individual inequalities. However, subtraction requires a bit more caution, especially in how it affects the direction and integrity of the inequalities. Let's clarify the situation with your example.
You start with two true inequalities:
$1 \leq 2 \leq 5$
$1 \leq 3 \leq 5$
Adding them is straightforward, and the result would indeed preserve the truth of the inequalities since addition is a direct operation that doesn't invert the direction of the inequalities.
However, when you "subtract" the inequalities, what you're effectively trying to do is not directly supported by the rules of inequalities. Let's break it down:- From $1 \leq 2$ and $1 \leq 3$, if you subtract the second inequality from the first, you're not actually performing a direct operation on the inequalities themselves. Instead, you might be thinking of subtracting the terms within each inequality, which isn't a valid operation in the context of manipulating inequalities directly.
The resulting expression you derived, $0 \leq 2 - 3 \leq 0$, which simplifies to $0 \leq -1 \leq 0$, is not true because $-1$ is not greater than or equal to $0$.
The confusion arises from attempting an operation (subtraction) that doesn't directly apply to inequalities the way addition or multiplication by a positive number does. The correct approach when dealing with inequalities and considering operations like subtraction is to work within the constraints of each inequality separately and avoid assuming that operations that apply to equations (like subtraction) will operate the same way on inequalities.