Ringschluss Explained

In mathematics, a Ringschluss is a mathematical proof technique where the equivalence of several statements can be proven without having to prove all pairwise equivalences directly.

In order to prove that the statements

are each pairwise equivalent, proofs are given for the implications , , , and .^{[1] [2]}

The pairwise equivalence of the statements then results from the transitivity of the material conditional.

Example

For

the proofs are given for , , and . The equivalence of and results from the chain of conclusions that are no longer explicitly given: .  . This leads to: .  . This leads to:

That is

.

Motivation

The technique saves writing effort above all. By dispensing with the formally necessary chain of conclusions, only

direct proofs need to be provided for instead of direct proofs. The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.

See also

• The term should not be confused with the invalid circular reasoning.

Notes and References

1. Book: Plaue . Matthias . Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis . Scherfner . Mike . 2019-02-11 . Springer-Verlag . 978-3-662-58352-4 . 26 . de . Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis.
2. Book: Struckmann . Werner . Mathematik für Informatiker: Grundlagen und Anwendungen . Wätjen . Dietmar . 2016-10-20 . Springer-Verlag . 978-3-662-49870-5 . 28 . de . Mathematics for Computer Scientists: Fundamentals and Applications.