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 path from back door to barn was covered by a grape-arbor, and every yard had its fruit trees.

Every time you transplant a pansy you cause its flowers to become smaller.

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 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.

1 ) Every age rewrites the events of its history in terms of what should have been, creating legends about itself that rationalize contemporary beliefs and excuse contemporary actions.

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

Every member of the family must have a vital place in its life.

Every telephone company, whether large or small, determines its own ANAC for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

* Every collecting institution must develop an emergency plan to protect its collections and train staff to carry it out.

* Every institution must assign responsibility for caring for collections to members of its staff.

Every ATM cell has an 8-or 12-bit Virtual Path Identifier ( VPI ) and 16-bit Virtual Channel Identifier ( VCI ) pair defined in its header.

" Every performance of its run was sold out.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

Every computer system provides at least one file-manager program for its native file system.

Every aspect of life was regulated to some degree by the party, and the will of its founding-president, Mobutu Sese Seko.

Every tile belongs to the two suits of its two values, e. g. 0 – 3 belongs both to the blank suit ( or 0 suit ) and to the 3 suit.

Every division ring is therefore a division algebra over its center.

Every field is, of course, one-dimensional over its center.

Every system must have a class designated as " root ", with one of its creation procedures designated as " root procedure ".

Every SE must be registered in the State where it has its registered office, in a register designated by the law of that State.

Every node on the Freenet network contributes storage space to hold files, and bandwidth that it uses to route requests from its peers.

The album's lead single, " She's Every Woman " peaked at number-one on the Billboard Country Chart, however its follow-up single, " The Fever " ( a cover of an Aerosmith song ) only peaked at number 23, becoming Brooks's first released Country single to not chart on the Top 10.

* Every small category has a skeleton.

Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters.

Every citation he found described an encounter between males where one party, the master, physically abused another, the slave.

Every person found infected was certified of the fact, removed to a hospital provided ( if his condition allow ), and kept under the orders of the medical officer.

Every time a coup was staged, some scapegoats or excuses were always found to justify it.

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space.

Every block, it seemed, became a parking lot while customers found it necessary to drive from one store to the next.

Every surveyed solid body in the solar system was found to be cratered, and there was no reason to believe that the Earth had somehow escaped bombardment from space.

Every animal and plant found elsewhere in Middle-earth exists in Valinor along with species endemic to Valinor.

Every Wednesday and Saturday during the growing season, the Lanesboro Farmers ' Market can be found in Sylvan Park.

Every boar hog being found to run after the above date the owner of such hog was to pay $ 1. 00 or forfeit the hog.

Every study has found that instead of people's MBTI scores clustering around two opposite poles, such as intuition vs. sensation, with few people scoring in the middle, people's scores actually cluster around the middle of each scale in a bell curve.

Every piece of scandal which could be raked together concerning Scaliger or his family is to be found there.

Every review noted the historical significance of the information, and those that noticed found the production values high, but the presentation of history mainly too dry, even resembling a lecture.

# Every word must be explained in all its meanings, in order that different interpretations may be found.

Every government, therefore, found him an useful servant ; and there was nothing in his opinions or in his character which could prevent him from serving any government.

" By the 1920s, copies of " Lift Every Voice and Sing " could be found in black churches across the country, often pasted into the hymnals.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.

Boselli and Mark Brunell helped found an Every Nation church in Jacksonville, Southpoint Community Church, and both are active in Champions for Christ.

Every June, Shionoe hosts a firefly festival to celebrate the arrival of the luminous insect which is found in the area.

Every kind of cuisine can be found here, and ranges from fine cuisine to hot dogs.

Every domain D can be embedded in an ω-complete domain completion ( called the ω-completion or simply the completion of D ) that is, in a precise sense, the smallest ω-complete domain containing D. The isolated elements of completion are precisely the isolated elements of D, but in general completion contains limit points that are not found in D. completion is uniquely determined up to isomorphism.

The duo's second long player, Every Man and Woman is a Star ( 1991 ) found critical acclaim, and Ultramarine was frequently seen at festivals such as Megadog and Glastonbury during the early to mid 1990s.

When Lucy buried the blanket, Snoopy took the time to dig for it himself ; and when he found it, Linus thanked him, upon which Snoopy thought " Every now and then I feel that my existence is justified!

Every such pipe or other means of conveyance, cock, vessel or utensil as mentioned, and all goods chargeable with a duty of excise or materials for the manufacture of such goods found therein, may be seized.