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

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

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 $$\{1,3\}$$.

• Which is bigger? The set of even integers or the set of odd integers? Letting $A$ be the even integers and $B$ the odd integers, note that $A+1 = B$ and $B+1 = A$. Commented Sep 17, 2020 at 17:12
• @JMoravitz Well I may have to constrain my discussion within finite sets Commented Sep 17, 2020 at 17:15
• @JMoravitz I made an edit such that the notation used is a weak order. I hope it is better now Commented Sep 17, 2020 at 17:21

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 $$$$A+B=\{a+b\,|\,a\in A\;\;\text{and}\;\;b\in B\}.$$$$ When one of those sets is just a singleton, we often write $$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.

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.