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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Applications for Sumsets?
Given an abelian group or semigroup $\mathcal{G}$ and sets $A,B\subseteq\mathcal{G}$, we define the sumset of $A$ and $B$ by $A+B=\lbrace a+b|a\in A,b\in B\rbrace$. We also denote by $hA=\lbrace a_1+.....
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Does $\overline{A+B}=\overline A+\overline B$ hold if $\overline A$ and $\overline B$ are compact?
Let us assume that we are working with subsets $A$, $B$ of some topological space $X$ such that also $A+B$ makes sense. (For example, we can have $X=\mathbb R^n$, $X$ could be a topological group, a ...
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On the definition of the Minkowski sum
Closed sets need not be mapped to closed sets by the Minkowski sum as the following example shows:
$$ S_1 := \{ (x, y) \mid x, y \in \Bbb R, x \geq 0, x y \geq 1 \},$$
$$ S_2 := {\Bbb R} \times \{0\} $...
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Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
Given two sets $S_1$ and $S_2$ in $\mathbb R^n$ define their sum by
$$S_1+S_2=\{x\in\mathbb R^n; x=x_1+x_2, x_1\in S_1, x_2\in S_2\}.$$
Prove that if $S_1$ and $S_2$ are compact, $S_1+S_2$ is ...
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The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$.
Define a set $C$ given by
$$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
Is $C$ a convex set?
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Is the Minkowski sum of two connected sets also connected? [closed]
Is the Minkowski sum of two connected sets, $A+B$, also connected?
Can I use translation mapping here?
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In how many different from a set of numbers can a fixed sum be achieved?
I have a set of number, and I want to know in how many ways from that set with each number being used zero, once or more times can a certain sum if at all, be achieved. The order doesn't matter. For ...
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An inequality on sumsets.
Given two finite sets $A,B\subset \mathbb{R}$, can we assert the inequality $$|A+B|^2\ge |A+A|\cdot|B+B|?$$
I tried to construct an injective function from $(A+A)\times (B+B)$ to $(A+B)^2$ but failed ...
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sigma notation for sumsets $\Sigma B$
I am reading a number theory paper about sumsets [1]. If $A, B \in \mathbb{Z}$ are two sets of integers we can define:
$$ A + B = \{ a + b : a \in A, b \in B \} \subseteq \mathbb{Z}$$
what do you ...
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The sum of a set $A$ with the empty set, $\varnothing$
Given that the sum of two sets is defined as
$$
A + B = \big\{ a + b : a \in A, b \in B \big\},
$$
how might one compute the sum
$$
A + \varnothing
$$
where $A$ may or may not be empty? In his book ...
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Limit measuring failure of sum-set cancellability
Suppose $A$, $B$ are finite sets of positive integers.
Let $$\mathcal{S}_n = \{C \subset [1,n] \, : \, A+C = B+C \}, $$ and denote $a_n = |\mathcal{S}_n|$.
Note that for any $X \in \mathcal{S}_n$ ...
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inclusion-exclusion principle
I don't have any way to see if I have done this in a correct way(no answers in my book). Did I do it right?
Question: "How many eight-bit strings either begin 100 or have the fourth bit 1 or both?"
...
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Proving $\sup \left( {A + B} \right) = \sup A + \sup B$ using the usual definition.
Prove $\sup \left( {A + B} \right) = \sup A + \sup B$ using the definition given in this problem (link). I was able to prove using lemma of the given definition. My attempt is here,
let $s = \sup A$,...
user405401
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Does $2X\neq X$ imply $2Y\neq Y$?
Let $n$ be a sufficiently large integer and $X\subseteq \mathbf{Z}_n$ a finite set of residues modulo $n$ such that $0\in X$ and $2X \neq X$. Fix also $y \in 2X\setminus X$ and define $Y=\{x \in X:y\...
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$p$ divides $ax+by+cz$
I am going to ask here a generalization of this other question:
Problem Fix $\varepsilon>0$ and let $p$ be a sufficiently large prime. Then, show that, for every $X\subseteq \{1,\ldots,p\}$ with $|...
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