It was all right to put a bunch of ranchers onto horses, to call them Night Riders, to set out to attack the largest mining combination the country had ever seen if all they wanted was adventure.

They recognized that slavery was a moral issue and not merely an economic interest, and that to recognize it explicitly in their Constitution would be in explosive contradiction to the concept of sovereignty they had set forth in the Declaration of 1776 that `` all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among them are life, liberty and the pursuit of happiness.

We of the liberal-led world got all set for peace and rehabilitation.

The Axioms required to make the theoretical machinery operate are set out tersely and powerfully, so that all permissible operations within the theory can be traced rigorously back to these axioms, rules, and primitive notions.

For by now the original cause of the quarrel, Philip's seizure of Gascony, was only one strand in the spider web of French interests that overlay all western Europe and that had been so well and closely spun that the lightest movement could set it trembling from one end to the other.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

In entering this union we will be surrendering most, if not all, of our economic autonomy to international bodies such as the Atlantic Institute ( recently set up ) or the O.E.C.D., I.M.F. and others.

The extent to which we can persuade the less developed countries to appraise their own resources, to set targets toward which they should be working, to establish in the light of this forward perspective the most urgent priorities for their immediate attention, and to do the other things which they must do to help themselves, all on a realistic long-term basis, will depend importantly on the incentives we place before them.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.

Above all, don't set up extravagant fringe benefits just to buy employee good will.

The huge market for changeable signs has spurred a universal demand for individual plastic letters, in all shapes and sizes -- and a number of companies are set up to supply them.

You can save all that shimming time if you set your windows in one, two, three order -- first the stud on one side, then the window, then the stud on the other side.

It was all set up so there would be no dust anywhere and so that their children would color in the coloring room, paint in the painting room, play with blocks in the block house, and do all the other things in the proper rooms at exactly the right time.

After all, if you were going to set up a workshop you had to have the proper equipment and that was that.

The set of all binomial distributions is called the family of binomial distributions, but in general discussions this expression is often shortened to `` the binomial distribution '', or even `` the binomial '' when the context is clear.

A unified set of regulations, applicable to all areas, was issued in 1929, and a complementary series of demographic inquiries in selected areas was instituted at the same time.

Unfortunately, however, and for reasons to be discussed in the following chapter, no rate relationships can be made completely nondiscriminatory as long as all or some of the rates must be set above marginal costs in order to yield adequate revenues.

J. Wheaton Smith, editor of the Warren Telegraph stated that `` the ends of justice must be satisfied, a solitary example must be set, in order that all those misnamed philantropists, who, actuated by a blind zeal, dare to instigate riot, treason, and murder, may heed it and shape their future course accordingly ''.

The set of all decisions is called the operating policy or, more simply, the policy.

The set of vectors Af constitutes the operating policy or, more briefly, the policy, and a policy is admissible if all the Af belong to S.

Houghton said 6 p.m. Friday had been set for a canvass of all 480 homes in the subdivision, which is located northeast of Dequindre and 14 Mile Road East.

The statement also points to a classic paradox: The more men turn toward God, who is not only in himself the paradigm of all unity but also the only ground on which human unity can ultimately be established, the more men splinter into groups and set themselves apart from one another.

: There is a set A such that for all functions f ( on the set of non-empty subsets of A ), there is a B such that f ( B ) does not lie in B.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

Similarly, the subset order ⊆ on the subsets of any given set is antisymmetric: given two sets A and B, if every element in A also is in B and every element in B is also in A, then A and B must contain all the same elements and therefore be equal:

For any set containing n elements, the number of distinct k-element subsets of it that can be formed ( the k-combinations of its elements ) is given by the binomial coefficient.

Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects ; more formally, the number of k-element subsets ( or k-combinations ) of an n-element set.

3-element subsets of a 5-element set

The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

The convex subsets of R ( the set of real numbers ) are simply the intervals of R.

# The union of a non-decreasing sequence of convex subsets is a convex set.

In mathematics, especially in order theory, the cofinality cf ( A ) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.

** In particular, let A be a set of size n, and consider the set of subsets of A containing no more than m elements.

Divisio is an exhaustive list of subsets of a set, in the sense that every member of the " divided " set is a member of one of the subsets.

An equivalence relation partition of a set | partitions a set into several disjoint set | disjoint subsets, called equivalence class es.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

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

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

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

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

* In set theory, an automorphism of a set X is an arbitrary permutation of the elements of X.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X