In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod ( the monoidal category of R-modules ).

This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if Y coincides with the codomain of ƒ.

* The statement that the binary operation ★ on a set S is idempotent, is equivalent to the statement that every element of S is idempotent for ★.

Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

Aristotle's assertion that "... it will not be possible to be and not to be the same thing ", which would be written in propositional logic as ¬ ( P ∧ ¬ P ), is a statement modern logicians could call the law of excluded middle ( P ∨ ¬ P ), as distribution of the negation of Aristotle's assertion makes them equivalent, regardless that the former claims that no statement is both true and false, while the latter requires that any statement is either true or false.

And in the self-referential spirit of the Liar Paradox, the phrase " it is true that ..." is equivalent to " this whole statement is true and ...".

In this case, the above definition can be characterized by the following equivalent statement:

The prime number theorem is equivalent to the statement that the nth prime number p < sub > n </ sub > is approximately equal to n ln ( n ), again with the relative error of this approximation approaching 0 as n approaches infinity.

The statement above, when is equivalent to

The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr.

According to the redundancy theory of truth, asserting that a statement is true is completely equivalent to asserting the statement itself.

then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.

In strict logical terms, via contraposition, this statement is equivalent to:

But since ( as above ) this statement is logically equivalent to ( 1 ) all ravens are black, it follows that the sight of a green apple is evidence supporting the notion that all ravens are black.

Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.

However, some significant differences between neoplatonism and Gregory's thought exist, such as Gregory's statement that beauty and goodness are equivalent, which contrasts with Plotinus ' view that they are two different qualities.

This statement came to be known as the mechanical equivalent of heat and was a precursory form of the first law of thermodynamics.

He mentions the beats only once '', when he refers to their having revived through mere power and abandonment and the unwillingness to, commit death in life some idea of a decent equivalent between verbal expression and actual experience,, but the entire narrative, is written in the tiresome vocabulary `` of '' that lost `` and '' dying cause, `` and in the '' `` sprung syntax that is supposed to supplant, our mother, tongue.

In the event the total of rupees accruing to the Government of the United States of America as a consequence of sales made pursuant to this Agreement is different from the rupee equivalent of $1,276 million, the amounts available for the purposes specified in paragraph 1, Article 2, will be adjusted proportionately.

In these readings, the double bass is either kept discreetly in the background, or it is dressed in clown's attire -- the musical equivalent of a bull in a china shop.

The cross-sectional area of the cylinders is determined and then the volume of the individual cylinders is computed by multiplying the area by the stroke length, which is the equivalent of the length of the cylinders.

It is well to bear in mind that gasoline will cost from $.80 to $.90 for the equivalent of a United States gallon and while you might prefer a familiar Ford, Chevrolet or even a Cadillac, which are available in some countries, it is probably wiser to choose the smaller European makes which average thirty, thirty-five and even forty miles to the gallon.

For example, when the film is only four minutes old, Neitzbohr refers to a small, Victorian piano stool as `` Wilhelmina '', and we are thereupon subjected to a flashback that informs us that this very piano stool was once used by an epileptic governess whose name, of course, was Doris ( the English equivalent, when passed through middle-Gaelic derivations, of Wilhelmina ).

It is convenient to classify a child's onset ages and completion ages as `` advanced '', `` moderate '' ( modal ), or `` delayed '' according to whether the child's age equivalent `` dots '' appeared to the left of, upon, or to the right of the appropriate short transverse line.

when it represents only itself and on which is its complement ( so that go on is semantically equivalent to board ), on has stronger stress than go does.

When I have instructions to leave is equivalent in meaning to I have instructions that I am to leave this place, dominant stress is ordinarily on leave.

When the same sequence is equivalent in meaning to I have instructions which I am to leave, dominant stress is ordinarily on instructions.

In the first of these sentences if by is the complement of come and Tuesday is an adjunct of time equivalent to on Tuesday, there will be strong stress on by in the spoken language ; ;

In the second sentence if drinking water is a gerundial clause and without drinking water is roughly equivalent in meaning to unless I drink water, there will be stronger stress on water than on drinking ; ;

but if drinking is a gerundial noun modifying water and without drinking water is equivalent to without water for drinking, there will be stronger stress on drinking than on water.

In the Steiners have busy lives without visiting relatives only context can indicate whether visiting relatives is equivalent in meaning to paying visits to relatives or to relatives who are visiting them, and in I looked up the number and I looked up the chimney only the meanings of number and chimney make it clear that up is syntactically a second complement in the first sentence and a preposition followed by its object in the second.

The mayor put the city under the municipal equivalent of martial law and declared a curfew.

xiii, " Cum ad Sacrosanctae Romanae Ecclesiae ") prescribes their work, determines how much they may charge for their labour, fixes a certain tax for an abstract or abridgment of twenty-five words, or their equivalent, 150 letters, forbids them to charge more, even though the abstract goes over twenty-five words but less than fifty words, enacts that the basis of the tax is the labour employed in writing, expediting, etc., the Bulls, and by no means the emoluments accruing to the recipient of the favour or benefice conferred by the Bull, and declares that whoever shall charge more than the tax fixed by him shall be suspended for six months from office, and upon a second violation of the law, shall be deprived of it altogether, and if the delinquent be an abbreviator, he shall be excommunicated.

He formulated the second law of electrolysis stating " the amounts of bodies which are equivalent to each other in their ordinary chemical action have equal quantities of electricity naturally associated with them.

Germany has constitutional guarantees against improper detention and have been implemented in statutory law in a manner that can be considered as equivalent to writs of habeas corpus.

In other words, a Miranda warning is a preventive criminal procedure rule that law enforcement is required to administer in order to protect an individual who is in custody and subject to direct questioning or its functional equivalent from a violation of his or her Fifth Amendment right against compelled self-incrimination.

* Measure of the Church of England is a law passed by the General Synod and the UK Parliament equivalent of an Act

The second law only applies to a particle that does not exchange matter with its surroundings, and so it is equivalent to write

In 984, the Persian mathematician Ibn Sahl wrote the treatise " On burning mirrors and lenses ", correctly describing a law of refraction equivalent to Snell's law.

Today, in all common law jurisdictions, standing mute is treated by the courts as equivalent to a plea of not guilty.

There are a number of equivalent statements of the theorem, which consists of two " supplements " and the reciprocity law:

Refraction is described by Snell's law, which states that for a given pair of media and a wave with a single frequency, the ratio of the sines of the angle of incidence θ < sub > 1 </ sub > and angle of refraction θ < sub > 2 </ sub > is equivalent to the ratio of phase velocities ( v < sub > 1 </ sub > / v < sub > 2 </ sub >) in the two media, or equivalently, to the opposite ratio of the indices of refraction ( n < sub > 2 </ sub > / n < sub > 1 </ sub >):

While it may seem unfair to forbid prosecution of crimes that law enforcement can later prove to a standard required by law (< em > cf .</ em >, e. g. beyond a reasonable doubt, clear and convincing evidence, and preponderance of the evidence ), the purpose of a statute of limitations or its equivalent is to ensure that the possibility of punishment for an act committed long ago cannot give rise to either a person's incarceration or the criminal justice system's activation.

Many common law jurisdictions have made similar statutory provisions, while a number of civil law jurisdictions have equivalent legislation incorporated into their civil codes.

The equivalent of tort in civil law jurisdictions is delict.

To tax ( from the Latin taxo ; " I estimate ") is to impose a financial charge or other levy upon a taxpayer ( an individual or legal entity ) by a state or the functional equivalent of a state such that failure to pay is punishable by law.

By Newton's law of universal gravitation and laws of motion, a body of mass m a distance R from the center of a sphere of mass M feels a force equivalent to an acceleration, where:

The waqf is an equivalent institution in Islamic law, restricted to charitable trusts.

Safe limits for mix of gases in trimix are generally accepted to be a maximum partial pressure of oxygen ( ppO < sub > 2 </ sub >— see Dalton's law ) of 1. 0 to 1. 6 bar and maximum equivalent narcotic depth of.

To maintain the tax advantage for income deferred into a 401 ( k ), the law stipulates the restriction that unless an exception applies, money must be kept in the plan or an equivalent tax deferred plan until the employee reaches 59½ years of age.