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In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
Then the relation x R y is equivalent with the equality x R = y R. It follows that equality is the finest equivalence relation on any set S in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element). In some contexts, equality is sharply distinguished from equivalence or ...
Congruence relation. 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. [1]
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [ 1 ] As an example, " is less than " is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the ...
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. [1][2] The principle is described by the physicist Albert Einstein 's formula: . [3] In a reference frame where the system is moving, its ...
In mathematics, Euler's identity[note 1] (also known as Euler's equation) is the equality where. is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for .
The congruence relation is an equivalence relation. The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
Energy–momentum relation. In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with ...