Therefore, the congruence of its secants, that is the image of a general plane field of lines, is of order Af and class Af.

For example, proposition I. 4, side-angle-side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well.

The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations.

There are relations which are both symmetric and antisymmetric ( equality and its subrelations, including, vacuously, the empty relation ), there are relations which are neither symmetric nor antisymmetric ( the relation " divides " on the set ℤ ; the relation " preys on " in biological sciences ), there are relations which are symmetric and not antisymmetric ( congruence modulo n ), and there are relations which are not symmetric but are antisymmetric (" is less than or equal to ").

* Recapitulation theory, a scientific theory influential on but no longer accepted in its original form by both evolutionary and developmental biology, namely, that the congruence in form between the same embryonic developmental stages of different species is evidence that the embryos are repeating the evolutionary stages of their ancestral history

For an algebra, given a congruence E on, the algebra is called the quotient algebra ( or factor algebra ) of modulo E. There is a natural homomorphism from to mapping every element to its equivalence class.

At its roots, geometry consists of a notion of congruence between different objects in a space.

One of them is that it introduced a completely new test for deciding whether Congress had exceeded its section-five powers: the " congruence and proportionality " test, a test that has proven to have great importance in the context of the Eleventh Amendment.

The " congruence and proportionality " requirement replaced the previous theory advanced in Katzenbach v. Morgan that the Equal Protection Clause is " a positive grant of legislative power authorizing Congress to exercise its discretion in determining the need for and nature of legislation to secure Fourteenth Amendment guarantees.

It is in terms of a Lie group G, to generalise the groups SL ( 2, R ) or PSL ( 2, R ) of modular forms, and a discrete group, to generalise the modular group, or one of its congruence subgroups.

An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and from V. 1 is impossible.

However, the majority opinion also stated that this part of the ADA lacked the " congruence and proportionality " required when Congress exercises its enforcement power under the Fourteenth Amendment, citing City of Boerne v. Flores.

The natural mapping from the integers to the congruence classes modulo n that takes an integer to its congruence class modulo n respects products.

There is not universal agreement on the successes and failures of SED, either in its congruence with standard theories of quantum mechanics, QED, and gravity, or in its compliance with observation.

Betweenness captures the affine aspect of Euclidean geometry ; congruence, its metric aspect.

The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish:

The invariant lines are the lines of the congruence of secants of **zg, since each of these meets Q in two points which are invariant.

The order of this congruence is Af, since Af secants of a curve of symbol ( B ) on a quadric surface pass through an arbitrary point.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

Each generator, **yl, of Af is also exceptional, for each is transformed into the entire congruence of secants of the curve into which that generator is transformed by the point involution on Q.

The class of the congruence is Af, since an arbitrary plane meets **zg in K points.

Finally, the image of a general bundle of lines is a congruence whose order is the order of the congruence of invariant lines, namely Af and whose class is the order of the image congruence of a general plane field of lines, namely Af.

If we take the congruence modulo for the right-hand-side expression, it is readily seen that

In number theory, a Carmichael number is a composite positive integer which satisfies the congruence

* A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure.

The algebraic properties ( 1-4 ) imply that equality is a congruence relation for a field ; in fact, it is essentially the only one.

Switching from combinatorics to algebra, both sides of this congruence are polynomials in a for each fixed value of b. The congruence therefore holds when a is any integer, positive or negative, provided that b is a fixed positive integer.

To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x < sup > 2 </ sup > + y < sup > 2 </ sup > = − 1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation

In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides.

The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal.

Zeller's congruence, a well-known algorithm for finding the day of week for any date, explicitly defines January and February as the " 13th " and " 14th " months of the previous year in order to take advantage of this regularity, but the month-dependent calculation is still very complicated for mental arithmetic:

PRI's The World is a global news radio, audio and multi-platform program created by Public Radio International based on the program's congruence with PRI's mission, and in order to fulfill the critical need for more original global news created for and provided to Americans.

Consider the timelike congruence generated by some timelike unit vector field X, which we should think of as a first order linear partial differential operator.

A line through two of these points, Af and Af, will be transformed into the entire bilinear congruence having the tangents to **zg at Af and Af as directrices.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class ( known as a congruence class ) as a single entity.

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by, is the set.

This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class or simply residue of the integer a, modulo n. When the modulus n is known from the context, that residue may also be denoted.

for all g < sub > 1 </ sub >, g < sub > 2 </ sub >, h < sub > 1 </ sub >, h < sub > 2 </ sub > ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.

For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

Equivalently, the primes are evenly distributed ( asymptotically ) among each congruence class modulo d.

( In other words, every congruence class except zero modulo p has a multiplicative inverse.

An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL ( n, Z ) we can reduce the entries to modular arithmetic in Z / NZ for any N > 1, which gives a homomorphism

Therefore, pp-waves belong to the Kundt class ( the class of Lorentzian manifolds admitting a null congruence with vanishing optical scalars ).

Since they constitute a very simple and natural class of Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other relativistic classical field theories of gravitation.

Some classes of solutions can be invariantly characterized using algebraic symmetries of the Weyl tensor: for example, the class of non-conformally flat null electrovacuum or null dust solutions admitting an expanding but nontwisting null congruence is precisely the class of Robinson / Trautmann spacetimes.