Concept

Congruence relation

Overview

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. I cannot see the answer to either of these contentions provided that you admit the materialistic theory of nature. With this theory nature at an instant in space is an independent fact. Thus we have to look for our preeminent congruence relation amid nature in instantaneous space; and Poincare is undoubtedly right in saying that nature on this hypothesis gives us no help in finding it. On the other hand Russell is…

   Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919

   Names Congruence relation several times; states what Congruence relation is.

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2. Any kind of dissimilarity is also of this kind: if the colour of A is unlike the colour of B, then the colour of B is unlike the colour of A. Relations of this sort are called _symmetrical_. Thus a relation is symmetrical if, whenever it holds between A and B, it also holds between B and A. All relations that are not symmetrical are called _non-symmetrical_. Thus "brother" is non-symmetrical, because, if A is a…

   Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy

   States what Congruence relation is.

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3. As an illustration of one possible logical system of geometry we may consider all relations of three terms which are analogous in certain formal respects to the relation "between" as it appears to be in actual space. A space is then defined by means of one such three-term relation. The points of the space are all the terms which have this relation to something or other, and their order in the space in question is…

   Bertrand Russell, Mysticism and Logic and Other Essays

   Close in meaning to Congruence relation; the term itself is not named.

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Related ideas

• Integer
• If and only if
• Logical conjunction
• Group (mathematics)
• Reflexive relation
• Set (mathematics)
• Subset
• Category theory
• Material conditional
• Equivalence relation

Place in the tradition

Part of First-order logic.

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