Difference between revisions of "Dedekind cut"
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A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. [[Real number|Real number]]). The concept of continuity of the real axis can be formulated in terms of Dedekind cuts of real numbers. | A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. [[Real number|Real number]]). The concept of continuity of the real axis can be formulated in terms of Dedekind cuts of real numbers. | ||
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)</TD></TR></table> | ||
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Revision as of 14:03, 18 October 2014
cut
A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. Real number). The concept of continuity of the real axis can be formulated in terms of Dedekind cuts of real numbers.
Comments
For the construction of $\mathbf R$ from $\mathbf Q$ using cuts see [a1].
References
| [a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |
How to Cite This Entry:
Dedekind cut. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_cut&oldid=31390
Dedekind cut. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_cut&oldid=31390
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article