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.

( 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.

For example, the inflected forms of a word can be represented, insofar as regular inflection allows, by a stem and a set of endings to be attached.

For the only time in the opera, words are not set according to their natural inflection ; ;

For a while Nick followed the twisting course of the bubbles, wondering which set came from Elaine.

" For some time, Lincoln continued earlier plans to set up colonies for the newly freed slaves.

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

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

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

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.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

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.

For much of the history of the series ( Volumes 4 through 29 ), settings in Gaul and abroad alternated, with even-numbered volumes set abroad and odd-numbered volumes set in Gaul, mostly in the village.

For the graphical user interfaces, AIX v2 came with the X10R3 and later the X10R4 and X11 versions of the X Window System from MIT, together with the Athena widget set.

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

For example the character set had no "↑" symbol, so exponentiation was an overstrike of "|" and "*".

For example, a Commodore 64 user calling an Atari BBS would use ASCII rather than the machine's native character set.

For the occasion on which everyone already knows everyone else and the host wishes them to meet one or a few honored newcomers, then the `` open house '' system is advantageous because the honored guests are fixed connective points and the drifting guests make and break connections at the door.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

For each of these lines meets Q in three points, namely two points on **zg and one point on one of the multiple secants.

For all these points he relies exclusively on Berle and Means's study of 1932, The Modern Corporation And Private Property.

: For a detailed presentation of the various points of view around the definition of " algorithm " see Algorithm characterizations.

For example, using Cartesian coordinates on the plane, the distance between two points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >) is defined by the formula

For two geometric objects P and Q represented by the relations P ( x, y ) and Q ( x, y ) the intersection is the collection of all points ( x, y ) which are in both relations.

For example, Barrett points out that the Pastoral Epistles are concerned that ministers of the generation of Timothy and Titus should pass on the doctrine they had received to the third generation.

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For purposes of scoring and reference, each player is identified by one of the points of the compass and thus North and South play against East and West.

For example, if the player rolls a 6 and a 3 ( notated as " 6-3 "), that player must move one checker six points forward, and another or the same checker three points forward.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For example, the file to which the link < tt >/ bin / ls </ tt > points in a typical Unix-like system probably has a defined size that seldom changed.

For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 4.

For a function of one variable, f, the set of all points ( x, y ) where y

For example, philosophy professor Robert Carroll points to Dianetics ' lack of empirical evidence:

For a set of scattered points in the plane, the diameter of the points is the same as the diameter of their convex hull.

For many cases, the boiling points of the components in the mixture will be sufficiently close that Raoult's law must be taken into consideration.

For the simple electric dipole given above, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges.

For the current loop, the magnetic dipole moment points through the loop ( according to the right hand grip rule ), with a magnitude equal to the current in the loop times the area of the loop.