Manipulating sets ($+$, etc). I was seeing a proof of the Open Mapping Theorem, in Kreyszig's book, and I have no problems with it. But there's a point in which he does something like: $$\begin{align}B_Y(0,r) \subset \frac{1}{a}(T(U) - Tx) &\implies aB_Y(0,r) \subset T(U)-Tx \\ &\implies B_Y(0, ra) \subset T(U) - Tx \\ &\implies Tx + B_Y(0,ra) \subset T(U) \\ &\implies B_Y(Tx, ra) \subset T(U).\end{align}$$Here $U$ is a set, $T$ is a bounded linear operator, $a > 0$, etc. I understand all of the steps, but this made me question:

In some situations, such as the above, we can do this sort of arithmetic with sets (and I don't mean set-algebra with $\cup$s and $\cap$s), for example, summing $Tx$ on both sides, cancelling, multiplying $a>0$ through, etc. Is there a pathologic situation where intuition fails when doing these calculations?


Obs: $A+B = \{a+b \mid a \in A, \,b \in B\}$, and $\alpha A = \{\alpha a \mid a \in A\}$.
 A: Here are a few things that fail:


*

*Distributive law: In general, $(\alpha + \beta)A \ne \alpha A + \beta A$.
Examples:


*

*$A=\mathbb N$, $\alpha=1$, $\beta=1$:
$(\alpha + \beta)A = 2\mathbb N = \{0,2,4,6,8,\ldots\}$
$\alpha A + \beta A = \mathbb N + \mathbb N = \mathbb N$

*$A=\mathbb N$, $\alpha=1$, $\beta=-1$
$(\alpha + \beta)A = 0\mathbb N = \{0\}$
$\alpha A + \beta A = \mathbb N + (-1)\mathbb N = \mathbb Z$


*Cancellation does not work. While $A = B \implies A+C = B+C$, the reverse is not true.
Example: $A = \{0\}$, $B = \{0, 1\}$, $C = \mathbb N$
Then $A + C = B + C = \mathbb N$, but $A \ne B$.
Actually, that's a corollary of the previous item, since cancellation is actually adding the negative to both sides.
What does work is cancelling a single term, for example $A + x = B + x \implies A = B$ (where $x$ is not a set, but a number or other arithmetic/algebraic entity).
As a rule of thumb, rules for arithmetic operations work as usual if only one set is involved in each arithmetic operation (note that repetitions of the same set count as different sets for this rule) but may fail if two or more sets are involved.
