Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

: Every non-empty set A contains an element B which is disjoint from A.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Every element of is a member of the equivalence class.

Every repetition of insertion sort removes an element from the input data, inserting it into the correct position in the already-sorted list, until no input elements remain.

Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms.

Every simple module is cyclic, that is it is generated by one element.

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

Every element has a successor ( there is no largest element ).

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

Every time a pixel on a triangle is rendered, the corresponding texel ( or texture element ) in the texture must be found.

Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).

Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by, where √ is called the radical sign or radix.

It is arguable that the recent Consultation Paper issued by CAFCASS and entitled Every Day Matters is a recognition of the hurt and resentment caused by the existing Family Law system, and is a call for radical change, particularly by the early adoption of Shared Residence in the vast majority of cases.

Every von Neumann regular ring has Jacobson radical

Every ring which is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.

Other radical movements flourished, such as the Townsend Plan for old age pension, and " Ham and Eggs ", which promised "$ 30 Every Thursday " to everyone over age 50.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

Every man in every one of these houses is a Night Rider.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Every infinite game in which is a Borel subset of Baire space is determined.

Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

* Every endomorphism of M is either nilpotent or invertible.

Every nilpotent group is p-nilpotent, and every p-nilpotent group is p-soluble.

Every so often the diminishing sound of a car came under the trailer as it slowed down for the wreck then speeded up again as it got clear.

Every so often he turned the knife.

Every man who dabbles in the market to make a little easy money on the side and suffers losses could at the time hardly face his wife who was wondering how her husband could be so dumb.

Every winter a kegful of this sauce was made and placed at the end of a row of four other kegs in the cellar, so that when its turn came, it was properly mellowed.

Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.

Every Andalusian region or comarca has its own variety of this popular food so there are many types of gazpacho.

Of Christianity he has this to say: "... it is not now true for me ... Every believing Christian is, I am sure, my spiritual brother ... but if systemically I called myself a Christian I feel that to most men I should imply too much and so tell a lie.

Every Mac made between 1986 and 1998 has a SCSI port on the back, making external expansion easy ; also, " toaster " Compact Macs did not have easily accessible hard drive bays ( or, in the case of the Mac Plus, any hard drive bay at all ), so on those models, external SCSI disks were the only reasonable option.

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.

King Mangra was so impressed that he allegedly remarked, " Every part of the Thai is blessed with venom.

Every objective has a different size ring, so for every objective another condenser setting has to be chosen.

Every associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions.

Ebert concludes his review with the following line: " Every time I see the film, I feel a great sadness, that a human imagination could be so limited that it sees its own extinction as a victory.

Every year, growers fold the higher parts of the plant downward so the plant stays at heights accessible by a standing human.

Every lethal viral disease presents a paradox: killing its host is obviously of no benefit to the virus, so how and why did it evolve to do so?

The convention details ( amongst other things ) that " Every driver shall keep to the edge of the carriageway appropriate to the direction of traffic ", and the " Drivers overtaking shall do so on the side opposite to that appropriate to the direction of traffic ", notwithstanding the presence or absence of oncoming traffic.

Every Duke thereafter has either died without heirs or succeeded to the Crown, and so has not been able to leave the Royal Family.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every autumn, according to Plutarch ( Life of Lycurgus, 28, 3 – 7 ), the Spartan ephors would pro forma declare war on the helot population so that any Spartan citizen could kill a helot without fear of blood or guilt ( crypteia ).

Every microwave oven sold has a protective interlock so that it cannot be run when the door is open or improperly latched.

Every ancient reference to Balaam considers him a non-Israelite, a prophet, and the son of Beor, though Beor is not so clearly identified.