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

A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).

Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.

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

Equivalently, a set is closed if and only if it contains all of its limit points.

More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.

Equivalently, a family of closed sets forms a base for the closed sets if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.

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, a generating set of a group is a subset such that every element of the group can be expressed as the combination ( under the group operation ) of finitely many elements of the subset and their inverses.

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, every dependent set contains a finite dependent set.

Equivalently, a function is convex if its epigraph ( the set of points on or above the graph of the function ) is a convex set.

Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its image.

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, the Planck mass is defined such that the gravitational potential energy between two masses m < sub > P </ sub > of separation r is equal to the energy of a photon ( or graviton ) of angular wavelength r ( see the Planck relation ), or that their ratio equals one.

Equivalently, an algebraic variety is projective if it can be embedded as a ( Zariski ) closed subvariety of P < sup > n </ sup >.

Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:

Equivalently, if P is a lattice, p ≠ top, and for all a, b in P,

Equivalently, x lies in an r-simplex with vertices in P, where.

Equivalently, C shatters A when the power set P ( A ) is the set

Equivalently, this formula can be written

Equivalently, using the exterior derivative, the curl can be expressed as:

Equivalently, we can think of tangent vectors as tangents to curves, and write

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

Equivalently, Player 1's strategy guarantees him a payoff of V regardless of Player 2's strategy, and similarly Player 2 can guarantee himself a payoff of − V.

Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit ( which can be calculated from the length of the input ) with polylogarithmic depth and a polynomial number of gates.

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, Ohm's law can be stated:

Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of X as the

Equivalently, we can think of " chunking " the search space into blocks, and then asking " in which block is the target item ?".

Equivalently, in polar coordinates ( r, θ ) it can be described by the equation

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.

Equivalently, if percents are used as whole numbers, as in 75 instead of 0. 75, the index can range up to 100 < sup > 2 </ sup >, or 10, 000.

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

Equivalently, the probability density function of the distribution can be written using the Dirac delta function as

Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

Equivalently, electron affinity can also be defined as the amount of energy required to detach an electron from a singly charged negative ion, i. e. the energy change for the process