For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

For all his successes, the nature of his proof stirred up more trouble than Hilbert could have imagined at the time.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

For the developers, Freeciv 1. 0 was a successful proof of concept, but a rather boring game, so they went back to XPilot.

For years the statue had been used by historians as proof of Hadrian's love of Hellenic culture.

Others have independently come to the same conclusion, and publication of independent analyses may be forthcoming [...] For scientists to label sight reports and questionable photographs as ' proof ' of such an extraordinary record is delving into ' faith-based ' ornithology and doing a disservice to science.

For example, the old instructions on burden of proof in civil cases read:

( For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.

For libertarians, that diversity of practice within a framework of common principles is proof of the vitality of those principles and of their flexibility and strength.

For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

( For example, the Coq proof assistant system features a tactic for Presburger arithmetic.

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

For instance, one report from Burma stated that red pandas were still fairly common in some areas, and was accompanied by a photograph of a " red panda " as proof.

For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity ( or the weaker property, truth ).

For contemporary Marxists, the existence of persistent unemployment is proof of the inability of capitalism to ensure full employment.

For the first three years of clad production, in lieu of proof sets, specimen sets were specially sold as " Special Mint Sets " minted at the San Francisco mint in 1965, 1966, and 1967 ( Deep Cameo versions of these spectacular coins are highly valued because of their rarity ).

“ For man, and especially the Proficient, is not the Couplement of Soul and body: the proof is that man can be disengaged from the body and disdain its nominal goods .” ( Enneads I. 4. 14 ) The human who has achieved happiness will not be bothered by sickness, discomfort, etc., as his focus is on the greatest things.

For a complete proof, see the reciprocity ( electromagnetism ) article.

For a proof of that, let's imagine two paths 1 and 2, both going from point A to point B.

For his part, Ernst Zermelo in his ( 1908 ) A new proof of the possibility of a well-ordering ( published at the same time he published " the first axiomatic set theory ") laid claim to prior discovery of the antinomy in Cantor's naive set theory.

For example, the credit card network does not require a personal identification number for authentication of the claimed identity ; and a small transaction usually does not even require a signature of the authenticated person for proof of authorization of the transaction.

For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.

For example, the word " Amerika " in German has a one-to-one equivalence to its meaning in modern English: it may denote North America, South America, or both, and in some instances refers to the United States only.

For example, the Cyrillic letter Р is usually written as R in the Latin script, although in many cases it is not as simple as a one-for-one equivalence.

The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences ( x < sub > n </ sub >)< sub > n </ sub > and ( y < sub > n </ sub >)< sub > n </ sub > in M, we may define their distance as

For instance, the equivalence relation ~ generated by:

For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

For such data, one must use a hash function that is compatible with the data equivalence criterion being used: that is, any two inputs that are considered equivalent must yield the same hash value.

For compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.

For a preorder "", a relation "<" can be defined as a < b if and only if ( a b and not b a ), or equivalently, using the equivalence relation introduced above, ( a b and not a ~ b ).

For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same ( here " the same " does not mean something like e. g. " the same up to translation and rotation ", but it means " exactly the same ").

For a strong acid and a strong base, the curve will be relatively smooth and very steep near the equivalence point.

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.

Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers ( which are not constructively equivalent to the Cauchy real numbers without countable choice < ref > For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman ; 1998 ; < cite > A weak countable choice principle </ cite >; available from .</ ref >).

For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions and are distinct as fractions, as different strings of symbols, but they " represent " the same rational number, the same point on a number line.

For equivalence, the impedance between any pair of terminals must be the same for both networks.

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.

For practical considerations in thermodynamics or engineering it is rarely necessary, nor convenient, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass.

For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass.

The table below lists rough equivalences with the year system in the rest of the United Kingdom ( For England and Wales, the equivalence given is for children born before 1st September ; the equivalence for those born from September to February for deferred pupils is given in brackets ):

Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant adjoint pair: For every scheme X and every commutative ring A we have a natural equivalence