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

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

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 a finite group, the derived series terminates in a perfect group, which may or may not be trivial.

For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.

For example, Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes ' number and Moser's number.

For some finite n-valued logics, there is an analogous law called the law of excluded n + 1th.

For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.

For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits.

For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas.

For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.

For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

* Family games like Roll Through the Ages, Birds on a Wire, or For Sale

For instance, Jim Devlin in 1876 pitched 66 complete games ( 662 innings pitched ) with a 1. 56 ERA but managed to record only five shutouts.

For 2012, the series will instead follow a 2-3 format with the higher seed playing the last 3 games ( as needed ) as home.

For months, speculation surrounded Selig and the possibility that he and Hank Aaron would not attend Bonds ' games as he closed in on the record.

For Hurricane Ike in 2008, Selig mandated that the Astros play two home games against the Chicago Cubs in his hometown of Milwaukee despite proximity to the visiting Cubs.

For the majority of games, there is no one set of universal rules by which the game is played, and the most common ruleset is no more or less than that.

For many reasons, including its age and its popularity among Western militaries, it is one of the most universally known card games in existence.

For this reason it is not hard to port games between these systems.

For the majority of the franchise's history the Cowboys played their home games at Texas Stadium.

For this reason the unit became a popular choice among those looking to pirate games.

For example, the computer baseball games Earl Weaver Baseball and Tony La Russa Baseball each had highly detailed simulations of the game strategies of those two baseball managers.

For the first 19 years of its existence, the team played its home games at Sun Life Stadium in Miami Gardens.

For Silius Italicus, who wrote as the games approached their peak, the degenerate Campanians had devised the very worst of precedents, which now threatened the moral fabric of Rome: " It was their custom to enliven their banquets with bloodshed and to combine with their feasting the horrid sight of armed men fighting ; often the combatants fell dead above the very cups of the revelers, and the tables were stained with streams of blood.

For example, the Storyteller System used in White Wolf Game Studio's storytelling games calls its GM the " storyteller ", while the rules-and setting-focused Marvel Super Heroes role-playing game calls its GM the " judge ".

For example, generally German-style games do not have a fixed number of players like chess or bridge ; though there is a sizable body of German-style games that are designed for exactly two players, most games can accommodate anywhere from two to six players ( with varying degrees of suitability ).

For this reason, the name " designer games " is often offered as a description of the genre.

For a few years, Activision continued to market Infocom's classic games in collections ( usually by genre, such as the Science Fiction collection ); in 1991, they published The Lost Treasures of Infocom, followed in 1992 by The Lost Treasures of Infocom II.

For example, the six button layout of the arcade games Street Fighter II or Mortal Kombat cannot be comfortably emulated on a console joypad, so licensed home arcade sticks for these games have been manufactured for home consoles and PCs.

For example, many flight simulators use it to switch the player's views, while other games sometimes use it as a substitute for the D-pad ; while computer gamepads modelled after PlayStation DualShock controllers assign POV switch scancodes to the D-pad of it.

In 1991 he was awarded the Balzan Prize for Genetics and Evolution " For his powerful analysis of evolutionary theory and of the role of sexual reproduction as a critical factor in evolution and in the survival of species ; for his mathematical models applying the theory of games to evolutionary problems " ( motivation of the Balzan General Prize Committee ).

For the last three seasons of his career, Bench caught only 13 games and played mostly first base or third base.

For their conference semi-finals, the Wiz beat the Dallas Burn in three games, winning the final game in a shootout.