Translation of a set: a set plus a vector $\{0,2\}+1=\{1,3\}$?

Question

I think the following operations make sense:

$\{0,2\}+1=\{1,3\}$

$[0,2]+1=[1,3]$

$\{(0,0),(2,1)\}+(1,0)=\{(1,0),(3,1)\}$

But, is it formally defined in any text? Has a mathematician defined that $\{0,2\}\leq\{1,3\}$ because $\{0,2\}+1=\{1,3\}$?

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Motivation:

We compare two sets: $A,B\in\mathbb R$; which one is greater? A common definition is called "Strong Set Order":

Define the binary relation $\leq_{s}$ as follows: $A \leq_{s} B \quad $ if for any $a \in A$ and $ b \in B$, $min\{a,b\} \in A$ and $max\{a,b\} \in B$

For instance, see the following picture, the first example $A\leq_s B$ while in the second case this is not true.

However, intuitively, it makes sense to also define that $\{0,2\}\leq\{1,3\}$ (although this is not true by the strong set order), because if we translate the set $\{0,2\}$ one unit to the right, then we get $\{1,3\}$.

Answer 1

But, is it formally defined in any text?

Yes, and it is called Minkowski summation. For two sets $A$ and $B$ in a vector space $V$, we write \begin{equation} A+B=\{a+b\,|\,a\in A\;\;\text{and}\;\;b\in B\}. \end{equation} When one of those sets is just a singleton, we often write $A+v$ as a shorthand for $A+\{v\}$ (where $v\in V$).

In regard to your questions on ordering, in general Minkowski addition will not preserve the same properties as the addition on $V$, e.g. the standard ordering on $\mathbb{R}$ is not necessarily preserved.

Answer 2

Answering your first question But, is it formally defined in any text?

The answer is positive. If $V$ is a vector space, $a$ is an element of $V$ and $U \subseteq V$ is a subset, you can define:

$$U+a = \{u + a \mid u \in U\}$$

$U+a$ is the translated of $U$ by the vector $a$.

$\{0,2\}$ is a subset of $\mathbb R$ and $\{1,3\}$ is its translated by the "vector" $1$. And similarly for your other two examples.