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

: 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 instance, because the 10 in the problem represents ten seconds, the expression f ′( 10 ) represents the change in pressure at a height of ten seconds, which is nonsense.

f and g are not known exactly: For example, the altitude where the car starts is not known and the temperature on the mountain is not known.

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

* For any homomorphism f: G → H, f () =.

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

For instance, if f ( x )

For instance, in the 55 Cancri system the first planet – 55 Cancri b – was discovered in 1996 ; two additional farther planets were simultaneously discovered in 2002 with the nearest to the star being named 55 Cancri c and the other 55 Cancri d ; a fourth planet was claimed ( its existence was later disputed ) in 2004 and named 55 Cancri e despite lying closer to the star than 55 Cancri b ; and the most recently discovered planet, in 2007, was named 55 Cancri f despite lying between 55 Cancri c and 55 Cancri d. As of April 2012 the highest letter in use is " j ", for the unconfirmed planet HD 10180 j ( HD 10180 h is the confirmed planet with the highest letter ).

For example, in Old and Middle English was an allophone of / f / occurring between vowels.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

For instance, German-speakers more often described, ( f .) " bridge " with words like ' beautiful ', ' elegant ', ' fragile ', ' peaceful ', ' pretty ', and ' slender ', whereas Spanish-speakers, which use puente ( m .) used terms like ' big ', ' dangerous ', ' long ', ' strong ', ' sturdy ', and ' towering '.

For instance, the precise definition for a homomorphism f to be iso is not only that it is bijective, and thus has an inverse f < sup >- 1 </ sup >, but also that this inverse is a homomorphism, too.

0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U. ( For a path γ from z < sub > 0 </ sub > to z lying entirely in U, define

For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley – Wiener theorem is an example of this.

For example, suppose we have a table like this, which gives some values of an unknown function f.

For example, the function for x ∈ Q satisfies f ( 0 )

A unary operation is idempotent if it maps each element of S to a fixed point of f. For a set with n elements there are

For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:

For every a ∈ A, there is λ ∈ C such that

For any x ∈ A,

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X ; a function ƒ with this property is called one-to-one, or information-preserving, or an injection.

For instance, for each n ∈ N, let

* For each a ∈ Σ ( a belongs to Σ ), the singleton language

For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X:

for all g < sub > 1 </ sub >, g < sub > 2 </ sub >, h < sub > 1 </ sub >, h < sub > 2 </ sub > ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.

For every dyadic fraction r ∈ ( 0, 1 ), we are going to construct an open subset U ( r ) of X such that:

: 3 ) For every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds:

For example, a smooth curve α ( t ): 1 → M has tangent vector α ′( t < sub > 0 </ sub >) in the tangent space TM ( α ( t < sub > 0 </ sub >)) at any point t < sub > 0 </ sub > ∈ ( 0, 1 ), and each such vector has length ‖ α ′( t < sub > 0 </ sub >)‖, where ‖·‖ denotes the norm induced by the inner product on TM ( α ( t < sub > 0 </ sub >)).

** For any i ∈

:: For any open neighborhood N of A, there is a positive constant T such that f ( t, b ) ∈ N for all real t > T.

For 1 < p, q < ∞ and f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), Hölder's inequality becomes an equality if and only if | f |< sup > p </ sup > and | g |< sup > q </ sup > are linearly dependent in L < sup > 1 </ sup >( μ ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f |< sup > p </ sup > = β | g |< sup > q </ sup > μ-almost everywhere.

# For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α.

For each weight λ ∈ h *, there exists a unique ( up to isomorphism ) simple highest-weight g-module with highest weight λ, which is denoted L ( λ ).

For example, if there is no inner model for a measurable cardinal, then the Dodd-Jensen core model, K < sup > DJ </ sup > is the core model and satisfies the Covering Property, that is for every uncountable set x of ordinals, there is y such that y ⊃ x, y has the same cardinality as x, and y ∈ K < sup > DJ </ sup >.

For every element a of a linearly ordered group G either a ∈ G < sub >+</ sub >, or − a ∈ G < sub >+</ sub >, or a = 0.

For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of $200 million.

For example, R. I. Pervo dates Acts to the first quarter of the 2nd century.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

From Comrade Semichastny's speech I learn that the government, ' would not put any obstacles in the way of my departure from the U. S. S. R .' For me this is impossible.

For example, an " X " is used to indicate a variable group amongst a class of compounds ( though usually a halogen ), while " R " is used for a radical, meaning a compound structure such as a hydrocarbon chain.

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

For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

For example, he believed ( as most Romans ) that his ancestor Appius Claudius Caecus had used the censorship to introduce the letter " R " and so used his own term to introduce his new letters.

For example, the Cyrillic letter Р is usually written as R in the Latin script, although in many cases it is not as simple as a one-for-one equivalence.

* For a more detailed account of Christine de Pizan ’ s rhetorical strategies refer to Jenny R. Redfern ’ s excerpt Christine de Pisan and The Treasure of the City of Ladies: A Medieval Rhetorician and Her Rhetoric ( in Reclaiming Rhetorica, ed.

For regions in R < sup > 3 </ sup > more complicated than this, the latter statement might be false ( see Poincaré lemma ).

For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

*( EF2 ) For all nonzero a and b in R,.

For example, an ester ( RCOOR ') has an ester functional group ( COOR ) and is composed of an alkoxy moiety (- OR ') and an acyl moiety ( RCO -), or, equivalently, it may be divided into carboxylate ( RCOO -) and alkyl (- R ') moieties.

For materials that absorb light, like metals and semiconductors, n is complex, and R < sub > p </ sub > does not generally go to zero.

For example, Latin P came to be written like Greek rho ( written Ρ but pronounced ), so the Roman letter equivalent to rho was modified to R to keep it distinct.

( For example, in spherical coordinates, let radius r go to R² / r where R is the Earth's radius.

For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.