Page "Axiom of choice" ¶ 19
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## Some Related Sentences

Every and set
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.
** Well-ordering theorem: Every set can be well-ordered.
** 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.
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
** Antichain principle: Every partially ordered set has a maximal antichain.
* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).
: Every non-empty set A contains an element B which is disjoint from A.
* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.
Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.
* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.
Every corporation, whether financial or union, as well as every division of the administration, were set up as branches of the party, the CEOs, Union leaders, and division directors being sworn-in as section presidents of the party.
Every DNS zone must be assigned a set of authoritative name servers that are installed in NS records in the parent zone, and should be installed ( to be authoritative records ) as self-referential NS records on the authoritative name servers.
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
# " Personality " Argument: this argument is based on a quote from Hegel: " Every man has the right to turn his will upon a thing or make the thing an object of his will, that is to say, to set aside the mere thing and recreate it as his own ".
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.
* Every singleton set
Every processor or processor family has its own machine code instruction set.
Every set is a class, no matter which foundation is chosen.
Every non-empty totally ordered set is directed.
* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

Every and has
Every soldier in the army has, somewhere, relatives who are close to starvation.
Every woman has had the experience of saying no when she meant yes, and saying yes when she meant no.
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 calculation has been made independently by two workers and checked by one of the editors.
Every retiring person has a different situation facing him.
Every family of Riviera Presbyterian Church has been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate.
Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.
`` Every woman in the block has tried that ''.
** Every surjective function has a right inverse.
** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
** Every vector space has a basis.
* Every small category has a skeleton.
** Every field has an algebraic closure.
** Every field extension has a transcendence basis.
** Every Tychonoff space has a Stone – Čech compactification.
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.
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 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 and choice
Every rational soul has naturally a good free-will, formed for the choice of what is good.
Every linguistic act involves choice, and choices are made on many scales.
Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).
Every other province had the freedom to regulate the religious question as it wished, although the Union stated every person should be free in the choice of his personal religion and no person should be prosecuted based on his or her religious choice.
Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.
Every year, the Knights and Setons raise money for a charity of their choice, their most successful fund raiser being the charity ball held every fall.
Every decision making process produces a final choice.
Every conceivable method of applying cutting fluid ( e. g., flooding, spraying, dripping, misting, brushing ) can be used, with the best choice depending on the application and the equipment available.
Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.
Every other year, they have a repertoire of student-directed plays, in which students apply to direct a play of their choice.
Dorfl speaks ' With The Beginning Of Every Word Being Capitalized ', as do all golems granted speech, and, like most golems, he is quite literal in his choice of words.
He wrote, " Every so often a director makes an inspiring casting choice to not hire a real actor for a role, but go with an unknown, an amateur.
Every now and then there was a crisis that called for military action, and Lower Fort Garry was the obvious choice for a base of operations.
Every effort is made to enable the preferred choice of course requested by students to be timetabled but where an option is not viable in terms of numbers and / or staffing, the Governors reserve the right to withdraw the subject option offer.
: Every independent person can acquire a domicile of choice by the jurisdiction of residence and intention of permanent or indefinite residence, but not otherwise.
The first paragraph of article 40, in the current Congolese constitution, states that " Every individual has the right to marry the person of their choice, of the opposite sex ".
Every Chinese New Years, the High School students get to spend a week in a location of their choice around the world.
Every year, hundreds of thousands of primary school students select to join Alliance High school as their first choice.

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