Difference between revisions of "Cauchy-davenport"
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− | The Cauchy-Davenport | + | The Cauchy-Davenport Theorem states that for all nonempty sets <math>A,B \subseteq \mathbb{Z}/p\mathbb{Z}</math> , we have that |
<cmath>|A+B| \geq \min\{|A|+|B|-1,p\},</cmath> | <cmath>|A+B| \geq \min\{|A|+|B|-1,p\},</cmath> | ||
where <math>A+B</math> is defined as the set of all <math>c \in \mathbb{Z}/p\mathbb{Z}</math> that can be expressed as <math>a+b</math> for <math>a \in A</math> and <math>b \in B</math>. | where <math>A+B</math> is defined as the set of all <math>c \in \mathbb{Z}/p\mathbb{Z}</math> that can be expressed as <math>a+b</math> for <math>a \in A</math> and <math>b \in B</math>. | ||
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+ | == Proof of the Cauchy-Davenport Theorem by the Combinatorial Nullstellensatz == | ||
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+ | == Proof by Induction == | ||
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+ | == Applications of the Cauchy-Davenport Theorem == |
Latest revision as of 13:39, 20 September 2019
The Cauchy-Davenport Theorem states that for all nonempty sets , we have that
where
is defined as the set of all
that can be expressed as
for
and
.