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 f is locally area-preserving, this non-overlapping property would make it area-preserving for all of S, so the area of f ( S ) would be the same as that of S, which is greater than 4.

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 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 every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~.

Since 2005 Apple Inc .' s Mac OS X operating system has come bundled with a dictionary application and widget which credits as its source " Oxford American Dictionaries ", and contains the full text of NOAD2.

# Since G is residually finite, if w is a word in the generators X of G then w ≠ 1 in G if and only of some mapping of X into S induces a homomorphism such that w ≠ 1 in S.

Since 2004, however, the X. Org Server, a fork of XFree86, has become predominant.

When he became pope, Leo X is reported to have said to his brother Giuliano: " Since God has given us the Papacy, let us enjoy it.

: Step 4: Since X assumes that Y is cooperating ; there must be another meaning to ( 2 ).

Since Mac OS X v10. 2, all controls are HIViews.

Since Mac OS X v10. 2, windows have a root HIView.

Since Mac OS X v10. 2, menus are HIObjects.

Since Mac OS X v10. 3, menu content may be drawn using HIViews, and all standard menus use HIViews to draw.

Since 1898, Perosi had been Director of the Sistine Chapel Choir, a title which Pius X upgraded to " Perpetual Director.

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.