: For any set X of nonempty sets, there exists a choice function f defined on X.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X.

One variation avoids the use of choice functions by, in effect, replacing each choice function with its range.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

Authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

With this alternate notion of choice function, the axiom of choice can be compactly stated as

: Every set has a choice function.

The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element ; this holds trivially.

The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on.

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.

The measure of combat efficiency in an indecisive campaign is a matter of personal choice.

The harder the choice, the more willing the league is to wade in.

If this choice is less exciting than New York Democrats may wish, it nevertheless must be made.

-- Her choice of one color means she is simply enjoying the motor act of coloring, without having reached the point of selecting suitable colors for different objects.

What it is trying to do is to protect the little man, too, as well as trying to maintain a flow of fresh meat to all stores, with choice of cut being made by the consumer, not the store.

The appointment of U Thant of Burma as the U.N.'s Acting Secretary General -- at this writing, the choice appears to be certain -- offers further proof that in politics it is more important to have no influential enemies than to have influential friends.

Only this time around the conditions are different and the choice is far harder.

It is by no stretch of the imagination a happy choice and the arguments against it as a practical strategy are formidable.

Its primary advantage is that it is a moral choice ; ;

but if he is up against China's crusading spirit in world affairs, he is going to be faced with the most agonizing choice in his life.

To the members of our Advisory Board, and most specially to its members who constitute our committees of selection, the Foundation is indebted for its successes of choice of Fellows.

The essence of contract is that one is free to make a choice of what one will or will not do.

Hence, the condition of freedom is a necessary condition for choice.

Advances in equipment and fabrication techniques give the sign or display manufacturer an extremely wide choice of production techniques, ranging from injection molding for intricate, smaller-size, mass-production signs ( generally 5000 units is the minimum ) to vacuum and pressure forming for larger signs of limited runs.

For outdoor signs and displays, where the problem of weathering resistance is no longer a factor, the choice of plastics is almost unlimited.

Since the choice is by lot each week, the outcomes of different trials are independent.

In early childhood the choice of a companion is likely to be for another child of his own age or a year or two older, who can do the things he likes to do ; ;

Here, the choice is that between the horns of a dilemma.

The fact seems to be that very many large branch stores are uneconomical, that the choice of location in the suburbs is as important as it was downtown, and that even highly suburbanized cities will support only so many big branches.

One of the greatest Homerists of our time, Frederick M. Combellack, argues that when it is assumed The Iliad and The Odyssey are oral poems, the postulated single redactor called Homer cannot be either credited with or denied originality in choice of phrasing.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

In the neighbourhood of x < sub > 0 </ sub >, for a the best possible choice is always f ( x < sub > 0 </ sub >), and for b the best possible choice is always f < nowiki >'</ nowiki >( x < sub > 0 </ sub >).

The domain A of f is called the search space or the choice set,

For the labial sounds f, v, b, p, and m, the only choice is to replace them with others.

The resulting definition, although it involves the choice of basis f, does not actually depend on f in an essential way.

The choice of basis f on the vector space V defines uniquely a set of coordinate functions on V, by means of

In particular, if v = e < sub > j </ sub > is the jth coordinate vector then ∂< sub > v </ sub > f is the partial derivative of f with respect to the jth coordinate function, i. e., ∂ f / ∂ x < sup > j </ sup >, where x < sup > 1 </ sup >, x < sup > 2 </ sup >, ... x < sup > n </ sup > are the coordinate functions on U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y < sup > 1 </ sup >, y < sup > 2 </ sup >, ... y < sup > n </ sup > are introduced, then

for all g, h, k in G. This c depends on the choice of the lift L, but a different choice of lift L ' ( g )= f ( g ) L ( g ) will result in a new cocycle

for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S ( with f = gj ), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point.

Next, the choice of pole ratio τ < sub > 1 </ sub >/ τ < sub > 2 </ sub > is related to the phase margin of the feedback amplifier .< ref > The gain margin of the amplifier cannot be found using a two-pole model, because gain margin requires determination of the frequency f < sub > 180 </ sub > where the gain flips sign, and this never happens in a two-pole system.

Not every number is an appropriate choice for the SNFS: you need to know in advance a polynomial f of appropriate degree ( the optimal degree is conjectured to be, which is 4, 5, or 6 for the sizes of N currently feasible to factorise ) with small coefficients, and a value x such that where N is the number to factorise.

Therefore the standard form of hypothesis H is that if Q defined as above has no fixed prime divisor, then all f < sub > i </ sub >( n ) will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials f < sub > i </ sub >( x ) with positive leading coefficients.

:: Because f ( x ) = e < sup > x </ sup > is guaranteed for rational x by the above properties ( see below ), one could also use monotonicity or other properties to enforce the choice of e < sup > x </ sup > for irrational x, but such alternatives appear to be uncommon.

for an appropriate choice of g and f we have the Abel – Plana formula: