Questions tagged [sumset]
For questions regarding sumsets such as $A+B$, the set of all sums of one element from $A$ and the other from $B$.
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How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
- 711
22
votes
4
answers
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Sum of closed and compact set in a TVS
I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
- 911
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votes
3
answers
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Is the Minkowski sum of a compact set and a closed set necessarily closed?
Why is it that if we have a compact set $X$ and a closed set $Y$ then the Minkowski sum $X+Y$ is necessarily closed?
Sorry for keep asking questions about the Minkowski sum, I am trying to figure out ...
- 287
15
votes
2
answers
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Measure of the Cantor set plus the Cantor set
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ ...
anonymous
4
votes
3
answers
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Example where closure of $A+B$ is different from sum of closures of $A$ and $B$
I need a counter example. I need two subsets $A, B$ of $\mathbb{R}^n$ so that $\text{Cl}(A+ B)$ is different of $\text{Cl}(A) + \text{Cl}(B)$, where $\text{Cl}(A)$ is the closure of $A$, and $A + B = \...
- 3,909
14
votes
3
answers
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The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$.
Define a set $C$ given by
$$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
Is $C$ a convex set?
- 151
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votes
3
answers
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Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
Given two sets $S_1$ and $S_2$ in $\mathbb R^n$ define their sum by
$$S_1+S_2=\{x\in\mathbb R^n; x=x_1+x_2, x_1\in S_1, x_2\in S_2\}.$$
Prove that if $S_1$ and $S_2$ are compact, $S_1+S_2$ is ...
- 201
7
votes
1
answer
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views
For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime
Inspired by Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n?, I wanted to first try to answer a simpler version of the problem, that considers only two ...
- 12.5k
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votes
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answer
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The sum of a set $A$ with the empty set, $\varnothing$
Given that the sum of two sets is defined as
$$
A + B = \big\{ a + b : a \in A, b \in B \big\},
$$
how might one compute the sum
$$
A + \varnothing
$$
where $A$ may or may not be empty? In his book ...
- 2,658
5
votes
1
answer
239
views
Prove there is a Bernstein set $B$ such that $B+B$ is also Bernstein
Show that there exists a Bernstein set $B$ such that $B+B$ is also Bernstein.
I have tried to use the definition that neither $B$ nor its complement contain a perfect set.
- 303
4
votes
1
answer
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views
Two uncountable subsets of real numbers without any interval and two relations
Are there two uncountable subsets $A, B$ of real numbers such that:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ?
We know that if one of them contains an interval,...
- 2,491
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votes
3
answers
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Sum of two open sets is open? [closed]
how to prove that if $A$ and $B$ are open in $(\mathbb{R},|.|)$ then $A+B$ is open ?
Where $A+B=\{a+b\mid a\in A,b\in B\}$
Thank you
- 5,193
3
votes
3
answers
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$\lim_{n\to\infty} \sum_{k=1}^n \frac{k!}{n!}$
I'm presented with the limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k!}{n!}$
I've got a hunch that it diverges to infinity but I wasn't able the prove that the sum is superior to a series diverging ...
- 115
1
vote
1
answer
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How do we know that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed? [closed]
The Minkowski sum of closed sets needn't be closed; $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ is the canonical example. However, its not clear to me how to prove this.
Question. How can we prove that $\...
- 68.1k
16
votes
1
answer
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views
Is the sum (difference) of Borel set with itself a Borel set?
Let $d \in \mathbb{N}$, $A \subset \mathbb{R}^d$, be a Borel set. Consider Minkowski sums
$$
\mathbb{S}(A) = A + A = \{x + y:\; x,y\in A \}
$$
$$
\mathbb{D}(A) = A - A = \{x - y:\; x,y\in A\}
$$
Must ...
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