Examples of using Nonempty in English and their translations into Serbian
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Cyrillic
Every nonempty subset of the natural numbers has a least element.
Recall that a continuum is a nonempty connected compact metric space.
Let a nonempty X⊆ N be given and assume X has no least element.
Repeat n times: Scan the array cells D,D,… until finding an i for which D is nonempty.
Every nonempty string that does not contain"+" or"=" and does not start with"0" is in L.
For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.
Product of a nonempty family of nonempty sets is nonempty.
The word enumerable is used because the following are equivalent for a nonempty subset B of the natural numbers.
Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set.
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
Next, consider any nonempty list I. Since I is nonempty, it has a head item, x, and a tail list, xs, so we can express it as(x: xs).
This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R.
Even if each of the Xi is nonempty, the Cartesian product may be empty if the axiom of choice(which is equivalent to the statement that every such product is nonempty) is not assumed.
This means that R is a linear order on X and every nonempty subset of X has an element which is minimal under R.
Second, since D must be nonempty, each i∈ I is adjacent to a vertex in D. Conversely, let D be a dominating set for G. Then it is possible to construct another dominating set X such that|X|≤|D| and X⊆ I: simply replace each u∈ D∩ U by a neighbour i∈ I of u.
The Cartesian product of any nonempty family of nonempty sets is nonempty.
Because 0 is the least element of N, it must be that 0∉ X. For any n∈ N, suppose for every k≤ n, k∉ X. Then S(n)∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n∈ N,n∉ X. Thus, X∩ N=∅, which contradicts X being a nonempty subset of N. Thus X has a least element.
For example, to show that the naturals are well-ordered-every nonempty subset of N has a least element-one can reason as follows.
If the set of all structures of a certain kind admits a well-founded partial order,then every nonempty subset must have a minimal element.
Moreover, the Banach fixed-point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f(x), f(f(x)), f(f(f(x))),… converges to the fixed point.
Define the boolean-valued function Q(i, s) to be the value(true or false)of"there is a nonempty subset of x1,…, xi which sums to s".
The sparsification lemma is proven by repeatedly finding large sets of clauses that have a nonempty common intersection in a given formula, and replacing the formula by two simpler formulas, one of which has each of these clauses replaced by their common intersection and the other of which has the intersection removed from each clause.
Let K{\displaystyle\scriptstyle K} be a set of indices and let( P k) k∈ K{\displaystyle\scriptstyle P_{ k}_{ k\ in K}}be a tuple(ordered collection) of nonempty subsets(the sites) in the space X{\displaystyle\scriptstyle X}.
The word enumerable is used because the following are equivalent for a nonempty subset B of the natural numbers: B is the domain of a computable function.
The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are well-ordered: every nonempty subset of the natural numbers has a unique least element under the natural ordering.
To find the minimum or maximum element, we scan from the beginning orend for the first nonempty bucket and find the minimum or maximum element in that bucket.