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 five years the cathedral carries out a major structural review.

Every year, each state has a state day at the cathedral, on which that state is recognized by name in the prayers.

Every year, on the third Friday in October, the Manchester Grammar School's Founders ' Day Service is held in the cathedral.

Every part of a Gothic cathedral is directly related to a “ core dimension ” which is used as an effort to achieve harmony and organic unity within the building where everything is linked rationally and proportionally, creating a coherent whole.

Every Eastern Orthodox church is usually dedicated to a saint, but the cathedral is currently only known for its dedication to the salvation of the Romanian people.

Every part of the cathedral was updated in the restoration ranging from expanded pews to better lighting with decorative painting on the interior walls and ceiling.

Every Orthodox temple has such a High Place even if it is not a cathedral.

Every Sunday he preached a sermon in Irish at the cathedral, and during his diocesan visitations he always addressed the people in their native tongue, which was still largely used in his diocese.

Every autumn however he came back to Utrecht to teach at the cathedral school.

* Every Day a Little Death — Countess Charlotte Malcolm and Anne Egerman

Every home game since May 15, 2003 has been sold out — a MLB record that has spanned almost nine years.

Every cleric in Holy Orders and every member of a religious order must publicly join in or privately read aloud ( i. e. using the lips as well as the eyes — it takes about two hours in this way ) the whole of the Breviary services allotted for each day.

Every information exchange between living organisms — i. e. transmission of signals that involve a living sender and receiver can be considered a form of communication ; and even primitive creatures such as corals are competent to communicate.

In his most popular book, Cultural Literacy — What Every American Needs To Know, he offers lists, quotations, and information regarding what he believes is essential knowledge.

" More immediately, though, he called for a massive build-up of U. S. arms production: " Every realist knows that the democratic way of life is at this moment being ' directly assailed in every part of the world … The need of the moment is that our actions and our policy should be devoted primarily — almost exclusively — to meeting this foreign peril.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

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.

* natural language — Every good boy deserves fruit ; and

Every subsequent round starts with the " strongest link "— the player with the most correct answers — from the previous round, unless that person has been voted off, in which case the second strongest answers first.

They draw three conclusions from Austin: ( 1 ) A performative utterance does not communicate information about an act second-hand — it is the act ; ( 2 ) Every aspect of language (" semantics, syntactics, or even phonematics ") functionally interacts with pragmatics ; ( 3 ) There is no distinction between language and speech.

Every monad arises from some adjunction — in fact, typically from many adjunctions — in the above fashion.

Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.

The Body of Liberties adopted in 1641 by the Massachusetts Bay colonists states, “ Every married woman shall be free from bodily correction or stripes by her husband, unless it be in his own defense from her assault .” In the United States, legal decisions in Mississippi ( 1824 ) and North Carolina ( 1868 and 1874 ) make reference to — and reject — an unnamed " old doctrine " or " ancient law " by which a man was allowed to beat his wife with a stick no wider than his thumb.

Every material has its own characteristic resistivity — for example rubber's resistivity is far larger than copper's.

Every character in the novel — both men and women — knows their place, and the traditional stereotypes of gender roles are repeated over and over again.

Every scene, every shot is the product of a keen and sensitive eye — an eye which has deep comprehension of the beauty of rugged people and a rugged world ".

He credited some of his success to his snowmobiling days: " Every winter, you would reckon on three or four big spills — and I'm talking about being thrown on to the ice at 100 miles per hour.

Every aspect of the methodology advocated in NKS is optimized to make experimentation as direct, easy, and meaningful as possible — while maximizing the chances that the experiment will do something unexpected.

Independence Day — Every year on July 4 Manteo holds an Independence Day celebration.

Every decision made by three of these " deputations " — and in each of them the lower clergy formed the majority — received ratification for the sake of form in general congregation, and if necessary led to decrees promulgated in session.