Equivalently, these statements are true in all models of ZFC but false in some models of ZF.

Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers.

Equivalently, a set is bounded if it is contained in some open ball of finite radius.

Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.

Equivalently X is arithmetical if X is or for some integer n. A set X is arithmetical in a set Y, denoted, if X is definable a some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in or for some integer n. A synonym for is: X is arithmetically reducible to Y.

If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:

Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, ... of elements of X such that x < sub > n + 1 </ sub > R x < sub > n </ sub > for every natural number n.

Equivalently, the extension E / F is Galois if and only if it is algebraic, and the field fixed by the automorphism group Aut ( E / F ) is precisely the base field F. ( See the article Galois group for definitions of some of these terms and some examples.

Equivalently, z < sub > 0 </ sub > is a ramification point if there exists a holomorphic function φ defined in a neighborhood of z < sub > 0 </ sub > such that ƒ ( z ) = φ ( z )( z − z < sub > 0 </ sub >)< sup > k </ sup > for some positive integer k > 1.

Equivalently ( because the Petersen graph has maximum degree three ) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges.

for some idempotent e in S. Equivalently,

Equivalently, they no longer depend on the Lie group G because there is nothing left to normalize, although there may still be some torsion.

Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem.

( Equivalently, many authors use the word " rotor " instead of " top ".

Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.

Equivalently, we can define a diagonal matrix as a matrix that is both upper-and lower-triangular.

Equivalently, we can define π < sub > n </ sub >( X ) to be the group of homotopy classes of maps g: < sup > n </ sup > → X from the n-cube to X that take the boundary of the n-cube to b.

The closure of the image of π ( A ) will be a C *- algebra of operators called the C *- enveloping algebra of A. Equivalently, we can define the

For a non-negative integer k, the kth Betti number b < sub > k </ sub >( X ) of the space X is defined as the rank of the abelian group H < sub > k </ sub >( X ), the kth homology group of X. Equivalently, one can define it as the vector space dimension of H < sub > k </ sub >( X ; Q ), since the homology group in this case is a vector space over Q.

Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.

Equivalently, we can define the Hadamard matrix by its ( k, n )- th entry by writing

Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a density function ρ such that on each coordinate patch there exist Darboux coordinates in which ρ is identically equal to one.

Equivalently, one can define Ricci rotation coefficients as follows:

Equivalently, we may take the open sets as a starting point and define

Equivalently, in differential form ( by the divergence theorem ):

Equivalently, given any sequence of elements of P

Equivalently, a problem is # P-complete if and only if it is in # P, and for any non-deterministic Turing machine (" NP machine "), the problem of computing its number of accepting paths can be reduced to this problem.

Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals ; that is, given any chain:

Equivalently ( and with no need to arbitrarily choose two points ) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points.

Equivalently, g ( n ) is the largest least common multiple ( lcm ) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence.

The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group C < sub > K </ sub > of K with respect to the natural topology on C < sub > K </ sub > related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism

Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Equivalently, any non-zero element of W has at least eight non-zero coordinates.

Equivalently, an action of a discrete group G on a topological space X is properly discontinuous if and only if any two points x and y have neighborhoods U < sub > x </ sub > and U < sub > y </ sub > such that there are only a finite number of group elements g with g ( U < sub > x </ sub >) meeting U < sub > y </ sub >.

Equivalently, a principal homogeneous space for a group G is a set X on which G acts freely and transitively, so that for any x, y in X there exists a unique g in G such that x · g = y where · denotes the ( right ) action of G on X.

Equivalently, linear polarization and TEM modes are mutually exclusive for a Gaussian beam or any other beam with curved wavefronts.

For a complex vector space V, it means for any x in C, C contains the disk with x on its boundary, centred on the origin, in the one-dimensional complex subspace generated by x. Equivalently, a balanced set is a circled cone.

Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.

Equivalently, in the density state formulation, for any ( valid ) density state ρ, we have

Equivalently, is conjugate to in if and only if and satisfy the Cauchy – Riemann equations in As an immediate consequence of the latter equivalent definition, if is any harmonic function on the function is conjugate to, for then the Cauchy – Riemann equations are just and the symmetry of the mixed second order derivatives, Therefore an harmonic function admits a conjugated harmonic function if and only if the holomorphic function has a primitive in in which case a conjugate of is, of course, So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain.

Equivalently, this thesis states that any function which has an algorithm is computable.

Equivalently the number of spanning trees is equal to the absolute value of any cofactor of the Laplacian matrix of G.

Equivalently, an edge is a bridge if and only if it is not contained in any cycle.

Equivalently, any LTI system can be characterized in the frequency domain by the system's transfer function, which is the Laplace transform of the system's impulse response ( or Z transform in the case of discrete-time systems ).