We have set up a central R & D department, as well as engineering-management departments -- about 80 people working on problems related to those of our plants.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

A binary relation is the special case of an n-ary relation R ⊆ A < sub > 1 </ sub > × … × A < sub > n </ sub >, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A < sub > j </ sub > of the relation.

The latter notation corresponds to viewing R as the characteristic function on " X " x " Y " for the set of pairs of G.

The case of R v Dudley and Stephens, in which two men were found guilty of murder for killing and eating a cabin boy while adrift at sea in a lifeboat, set the precedent that necessity is no defence to a charge of murder.

* R carrying the lower limit topology satisfies the property that no uncountable set is compact.

* In the cocountable topology on R ( or any uncountable set for that matter ), no infinite set is compact.

is a function defined on a subset I of the set R of real numbers.

This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval

The convex subsets of R ( the set of real numbers ) are simply the intervals of R.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

* If x < sub > 0 </ sub > is a real number, we can turn the set R −

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".

* The empty relation R on a non-empty set X ( i. e. aRb is never true ) is vacuously symmetric and transitive, but not reflexive.

For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

It was all right to put a bunch of ranchers onto horses, to call them Night Riders, to set out to attack the largest mining combination the country had ever seen if all they wanted was adventure.

They recognized that slavery was a moral issue and not merely an economic interest, and that to recognize it explicitly in their Constitution would be in explosive contradiction to the concept of sovereignty they had set forth in the Declaration of 1776 that `` all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among them are life, liberty and the pursuit of happiness.

We of the liberal-led world got all set for peace and rehabilitation.

The Axioms required to make the theoretical machinery operate are set out tersely and powerfully, so that all permissible operations within the theory can be traced rigorously back to these axioms, rules, and primitive notions.

For by now the original cause of the quarrel, Philip's seizure of Gascony, was only one strand in the spider web of French interests that overlay all western Europe and that had been so well and closely spun that the lightest movement could set it trembling from one end to the other.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

In entering this union we will be surrendering most, if not all, of our economic autonomy to international bodies such as the Atlantic Institute ( recently set up ) or the O.E.C.D., I.M.F. and others.

The extent to which we can persuade the less developed countries to appraise their own resources, to set targets toward which they should be working, to establish in the light of this forward perspective the most urgent priorities for their immediate attention, and to do the other things which they must do to help themselves, all on a realistic long-term basis, will depend importantly on the incentives we place before them.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.

Above all, don't set up extravagant fringe benefits just to buy employee good will.

The huge market for changeable signs has spurred a universal demand for individual plastic letters, in all shapes and sizes -- and a number of companies are set up to supply them.

You can save all that shimming time if you set your windows in one, two, three order -- first the stud on one side, then the window, then the stud on the other side.

It was all set up so there would be no dust anywhere and so that their children would color in the coloring room, paint in the painting room, play with blocks in the block house, and do all the other things in the proper rooms at exactly the right time.

After all, if you were going to set up a workshop you had to have the proper equipment and that was that.

The set of all binomial distributions is called the family of binomial distributions, but in general discussions this expression is often shortened to `` the binomial distribution '', or even `` the binomial '' when the context is clear.

A unified set of regulations, applicable to all areas, was issued in 1929, and a complementary series of demographic inquiries in selected areas was instituted at the same time.

Unfortunately, however, and for reasons to be discussed in the following chapter, no rate relationships can be made completely nondiscriminatory as long as all or some of the rates must be set above marginal costs in order to yield adequate revenues.

J. Wheaton Smith, editor of the Warren Telegraph stated that `` the ends of justice must be satisfied, a solitary example must be set, in order that all those misnamed philantropists, who, actuated by a blind zeal, dare to instigate riot, treason, and murder, may heed it and shape their future course accordingly ''.

The set of all decisions is called the operating policy or, more simply, the policy.

The set of vectors Af constitutes the operating policy or, more briefly, the policy, and a policy is admissible if all the Af belong to S.

Houghton said 6 p.m. Friday had been set for a canvass of all 480 homes in the subdivision, which is located northeast of Dequindre and 14 Mile Road East.

The statement also points to a classic paradox: The more men turn toward God, who is not only in himself the paradigm of all unity but also the only ground on which human unity can ultimately be established, the more men splinter into groups and set themselves apart from one another.

But the real beginnings of this development in him go back to the opposing of grammar school, and probably if it had not been this occasion and these Latin lines it would have been some others, such as the first prolusion, that set off this streak in him of unbridled and scathing verbal attack on an enemy.

After the usual Honorable Sirs, it went on to say that there had been set off to the widow one full third part of the real estate of the deceased Salu Norberg, one lower room, on the Western side, privileges to the well and bake-oven and to one third of the cellar ( I can show you the cellar when we go up ), also one Cow Right, and lastly they set off to the widow her own land that she brought with her as dower, namely the Beech Pasture.

The tragic irony of the play is that the very belief in and concern with a devil who could be met in the woods and combatted with formulae set out in books was the very thing that prevented them from detecting the real devil when he came among them.

This did not set Gershwin back, as his real intent abroad was to complete a new work based on Paris and perhaps a second rhapsody for piano and orchestra.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers.

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

" Amazing Grace " set to " New Britain " was included in two hymnals distributed to soldiers and with death so real and imminent, religious services in the military became commonplace.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

* Since Big Brother 2, the UK series always opens with a twist which have included the public being able to choose the final housemate out of three possibilities ( Big Brother 2 ), the public voting for a housemate to leave during the first week and then the housemates choosing between the two housemates with the least number of votes ( Big Brother 3 ), First Night Nominations ( Big Brother 4 ), Suitcase Nominations ( Big Brother 5 ), Unlucky Housemate 13 ( Big Brother 6 ), Big Brother Hood ( Big Brother 7 ), an all-female House and the first inclusion of twins as contestants ( Big Brother 8 ), the first couple to enter as housemates and set a secret task to hide their real relationship ( Big Brother 9 ), all " housemates " really being " non-housemates " who had to earn their housemate status ( Big Brother 10 ), a mole entering the House with an " Impossible Task " ( Big Brother 11 ), Jackie Stallone entering a house containing her son's ex-wife ( Celebrity Big Brother 3 ), the entrance of a non-celebrity in a celebrity edition ( Celebrity Big Brother 4 ) and Jade Goody's family announced to be visiting.

* The set of real ( or complex ) numbers is a Banach algebra with norm given by the absolute value.

* The set of all real or complex n-by-n matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.

: There is no set whose cardinality is strictly between that of the integers and that of the real numbers.