If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

If the anchor continues to drag, or sets after having dragged too far, it should be retrieved and moved back to the desired position ( or another location chosen.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

If a player wins two sets, or gets a win and a tie, that player wins the game.

If each player wins a set, or both sets end tied, there is a 3-end tiebreaker to determine a winner.

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If the sets are short to medium length the caller will often try to run the dance until each couple has danced with every other couple both as a 1 and a 2 and returned to where they started.

If two sets are used, each player controls two differently colored sets of pieces at opposite corners of the star.

If two sets are played, the pieces can either go into the opponent's starting corners, or one of the players ' two sets can go into an opposite empty corner.

If three sets are played, the pieces usually go into the opponent's starting corners.

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

If this precaution was not taken, the potential customers would be at the mercy of " 2 sets of carob seeds ".

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

If the two sets of bodies do not have concurrent jurisdiction but, as in the case of the International Criminal Court ( ICC ), the relationship is expressly based on the principle of complementarity, i. e. the international court is subsidiary or complementary to national courts, the difficulty is avoided.

# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.

A measure μ is monotonic: If E < sub > 1 </ sub > and E < sub > 2 </ sub > are measurable sets with E < sub > 1 </ sub > ⊆ E < sub > 2 </ sub > then

A measure μ is countably subadditive: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … is a countable sequence of sets in Σ, not necessarily disjoint, then

A measure μ is continuous from below: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n </ sub > is a subset of E < sub > n + 1 </ sub > for all n, then the union of the sets E < sub > n </ sub > is measurable, and

If it continues indefinitely it is nearly a statistical certainty that a mistake will be made and that the devastation will begin ''.

If the list is a statistical population, then the mean of that population is called a population mean.

If the list is a statistical sample, we call the resulting statistic a sample mean.

If empirical data reach significance under the appropriate statistical formula, the research hypothesis is supported.

If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the " sample mean " () to distinguish it from the " population mean " ( or < sub > x </ sub >).

If the position was found to be r < sub > 0 </ sub > then in an interpretation satisfying CFD, the statistical population describing position and momentum would contain all pairs ( r < sub > 0 </ sub >, p ) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair ( r < sub > 0 </ sub >,⊥) where ⊥ denotes an undefined value.

: Q: If three hens lay three eggs in three days, how many eggs does a ( statistical ) hen lay in one day?

If, instead, the hypothesis was specified after some of the data were examined, and specifically tuned to match the direction in which the early data appeared to point, the calculation would overestimate statistical significance.

If one corrects and adjusts for the confounding by smoking, via stratification or multivariable regression modelling to statistical control for smoking, then one finds that the association between birth weight and mortality may be attenuated towards the null.

If there is a large enough number of atoms, statistical temperature can be estimated from the instantaneous temperature, which is found by equating the kinetic energy of the system to nk < sub > B </ sub > T / 2 where n is the number of degrees of freedom of the system.

If sample size is under a researcher's control, a good choice depends on the statistical power of the test, the effect size that the test must reveal and the desired significance level.

If so, what does statistical mechanics tell us about the arrow of time?

If we assume that a noisy signal or image is noiseless when we try to make a statistical estimate of g, our estimate will be incorrect.

If we assume that each possible chain conformation has an equal statistical weight, it can be shown that the probability P ( r ) of a polymer chain in the population to have distance r between the ends will obey a characteristic distribution described by the formula

If that is true then statistical analysis also cannot say anything useful about any performance that depends on customers or on other accumulating asset-stocks – which is always the case.

If equality of opportunity is violated, perhaps by discrimination which affects a subgroup or population over time, it is possible to make this determination using statistical analysis, but there are numerous difficulties involved.

If the required commercial invoice is needed for statistical purposes, the importer has to produce the commercial invoice within 50 days from the date Customs releases the goods to the importer.

The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used: If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is no longer a good approximation.

If the summed responses fulfill these assumptions, parametric statistical tests such as the analysis of variance can be applied.

If the calculated p-value is below the threshold chosen for statistical significance ( usually the 0. 10, the 0. 05, or 0. 01 level ), then the null hypothesis is rejected in favor of the alternative hypothesis.

If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found.

In mathematics, the Sato – Tate conjecture is a statistical statement about the family of elliptic curves E < sub > p </ sub > over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If N < sub > p </ sub > denotes the number of points on E < sub > p </ sub > and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for N < sub > p </ sub >.

If the program is fairly large, and there are enough data, statistical analysis can be used to make a reasonable case for the program by showing, for example, that other causes are unlikely.

If " econophysics " is taken to denote the principle of applying statistical mechanics to economic analysis, as opposed to a particular literature or network, priority of innovation is probably due to Farjoun and Machover ( 1983 ).