A set S is called countable if there exists an injective function f from S to the natural numbers Since there is an obvious bijection between and it makes no difference whether one considers 0 to be a natural number of not.

Since the possible values of f are well-ordered, this property can be established by proving for any r ( not in I ) with minimal value of f ( r ) in its class.

Since the stopping force F times that distance must be equal to the head's kinetic energy, it follows that F will be much greater than the original driving force f — roughly, by a factor D / d.

Since < ƒ, ƒ > ≥ 0, φ ( f * f ) ≥ 0 for all ƒ in L < sup > 2 </ sup >( m ), where ƒ * is pointwise conjugate of ƒ.

Since the focal length f of a lens is dependent on the refractive index n, different wavelengths of light will be focused on different positions.

Since the topology on R < nowiki ></ nowiki > X < nowiki ></ nowiki > is the ( X )- adic topology and R < nowiki ></ nowiki > X < nowiki ></ nowiki > is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients ( so that they belong to the ideal ( X )): f ( 0 ), f ( X < sup > 2 </ sup >− X ) and f ( ( 1 − X )< sup >− 1 </ sup > − 1 ) are all well defined for any formal power series f ∈ R < nowiki ></ nowiki > X < nowiki ></ nowiki >.

Since equalizers may be constructed from pullbacks and binary products ( consider the pullback of ( f, g ) along the diagonal Δ ), a category is complete if and only if it has pullbacks and products.

* Since m is injective, f ( a ) =

* Since the composition of g and f is trivial, c =

Since the gradient of a function is perpendicular to the contour lines, this is the same as saying that the gradients of f and g are parallel.

Since x is now fixed, f ( x ) is a fixed countable set and has Lebesgue measure zero.

* Since we can predict with virtual certainty that " y is not in f ( x )" after the first dart is thrown, and since this prediction is valid no matter what the first dart does, we should be able to make this prediction before the first dart is thrown.

* Since " y is not in f ( x )" is predictably true, by the symmetry of the order in which the darts were thrown ( hence the name " axiom of symmetry ") we should also be able to predict with virtual certainty that " x is not in f ( y )".

Since f ( X ) is irreducible, ( f ( X )) is a maximal ideal and hence is, in fact, a field.

Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least ( or greatest ) fixed point.

The set of ( p < sub > i </ sub >, n < sub > i </ sub >) is thus a sampling of a continuous vector field V. The implicit function f is found by integrating the vector field V. Since not every vector field is the gradient of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field V to be the gradient of a function f is that the curl of V must be identically zero.

Since the Supreme Court's decision of that year this is more doubtful ; ;

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Since it is not far from Viareggio, he will visit Puccini's house, as he never fails to do, to pay his respects to the memory of the composer of La Boheme, which he considers one of Puccini's masterpieces.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

Since the air is a continuum, the network of communication remains intact regardless of the positions or motions of the points ( the people ) in the net.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the difficulty of drawing the net is great, we will merely discuss it.

Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.

Since civilizational change is the most difficult to perceive and analyze, it seldom is given adequate attention.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since more is known about Quiney than about any other acquaintance of Shakespeare in Stratford, his career may be followed to its sudden end in 1602.

Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.

Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Since the work is done by hand, the only limitation, it is said, `` is that of human conception ''.

Since the apparatus is new, it requires experimentation and changes in technique.

Since the obvious is not always true, the Republican National Committee wisely analyzed its defeat of last autumn and finds that it occurred, as suspected, in the larger cities.

Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.

Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.

Since the goal of our international planners is a World Government, this Atlantic Community would mark a giant step in that direction for, once American economic autonomy is absorbed, a larger grouping is a question of time.

Since its reformation under private law in 1994, Deutsche Bahn AG ( DB AG ) no longer publishes details of the tracks it owns ; in addition to the DBAG system there are about 280 privately or locally owned railway companies which own an approximate 3, 000 km to 4, 000 km of the total tracks and use DB tracks in open access.

Since northeastern Pennsylvania was a rich source of anthracite coal, anthracite-fired furnaces using locally available iron ores were built throughout eastern Pennsylvania, helping to make the state a leader in iron production in the latter half of the 19th century.

Since 2000, export vehicles have been offered with thermal imaging night-vision gear of French manufacture as well ( though it may be more likely that they might simply use the locally manufactured ' Buran-Catherine ' system, which incorporates a French thermal imager ).

Since methamphetamine is easy to produce, the substance is manufactured locally in staggering quantities.

Since many of the first issues of these stamp issuing states were printed locally using primitive methods ( generally typographed ), they can be very rare.

Since the problem is not locally apparent in the MMX code, the bug can be very time consuming to find and correct.

Since version 8i, the Oracle RDBMS also supports " locally managed " tablespaces which can store space management information in bitmaps in their own headers rather than in the tablespace ( as happens with the default " dictionary-managed " tablespaces ).

Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps.

Since attendance at a temple sealing is restricted, a couple will be married locally by a person who is duly authorized to perform marriages.

Since the 1970s, there has been a flowering of musical diversity, with Israeli rock, folk and jazz musicians creating and performing extensively, both locally and abroad.

Since the 16th century, Baldock has been a centre for malting, subsequently becoming a regional brewing centre with at least three large brewers still operating at the end of the 19th Century, despite a decline in demand for the types of beer produced locally.

Since other rifts are known to occur in North America's stress environment but not all are associated with modern earthquakes, ( for example the Midcontinent Rift System that stretches from Minnesota to Kansas ), other processes could be at work to locally increase mechanical stress on the New Madrid faults.

Since the login page itself must be presented to the client, either that login page is locally stored in the gateway, or the web server hosting that page must be " whitelisted " via a walled garden to bypass the authentication process.

* SIAI-Marchetti S. 211 — Since 1984, 32 × S. 211s were acquired for RSAF's Basic Jet Training ( BJT ) program ( this figure includes 24 airframes which were assembled locally by Singapore Aircraft Industries plus two former Haitian aircraft acquired as attrition replacements in 1994 ).

Since 1989 there is a locally elected community board with limited, mainly representational powers, in line with other neighbourhoods in Auckland City.

Since M-39 was moved to Southfield Road in the end of the 1950s, Schaefer Highway has been a locally maintained road.

Winchell, in his 1914 report, Mining Districts of the Dillon Quadrangle, referred to the highest peak in the Tobacco Roots as “ Jefferson Peak, locally called Old Hollowtop .” Since that time the more northerly, and higher, of two adjacent peaks has usually been called Hollowtop.

Since other rifts are known to occur in North America's stress environment but not all are associated with modern earthquakes, ( for example the Midcontinent Rift System that stretches from Minnesota to Kansas ), other processes could be at work to locally increase mechanical stress on the New Madrid faults.

Since the mail was stored locally, users with laptop computers would be able to read and compose mail while on the road.

Since 2008, re-runs have begun screening on The Comedy Channel as part of the channel's " Aussie Gold " block of locally made, classic comedy programming.

Since 1982 WABC has programmed a talk radio format, and has been one of the most successful talk stations locally and in the United States.

Since the mid-19th century, Brockworth has been known locally for the annual rolling of Double Gloucester cheese down Cooper's Hill.

Since 1981 the Committee has organized an annual dinner with a guest of honour from Oxford University, who has also given one or more lectures locally in Oslo.

Since then, the award has been given to locally manufactured vehicles on 11 out of 28 occasions.