In mathematics, the lexicographic or lexicographical order, ( also known as lexical order, dictionary order, alphabetical order or lexicographic ( al ) product ), is a generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters.

Artists and sculptors tried to find this ideal order in relation with mathematics, but they believed that this ideal order revealed itself not so much to the dispassionate intellect, as to the whole sentient self.

* Atom ( order theory ) in mathematics

It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them.

In mathematics a combination is a way of selecting several things out of a larger group, where ( unlike permutations ) order does not matter.

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.

Much of mathematics is grounded in the study of equivalences, and order relations.

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property that is preserved by an isomorphism and that is true of one of the objects, is also true of the other.

The German philosopher Arthur Schopenhauer designates this " inner nature " with the term Will, and suggests that science and mathematics are unable to penetrate beyond the relationship between one thing and another in order to explain the " inner nature " of the thing itself, independent of any external causal relationships with other " things ".

In mathematics, especially in order theory, a preorder or quasi-order is a binary relation that is reflexive and transitive.

This change from a quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i. e. functions of functions: " We can decide that mathematics is to confine itself to functions of functions which obey the above assumption " ( PM 2nd Edition p. 401, Appendix C ).

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order.

In both mathematics and the relational database model, a set is an unordered collection of unique, non-duplicated items, although some DBMSs impose an order to their data.

In mathematics, a tuple has an order, and allows for duplication.

The players of the game may not need to use mathematics in order to play mathematical games.

For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.

Mathematical puzzles require mathematics in order to solve them.

In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements.

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

In mathematics, a well-order relation ( or well-ordering ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

In mathematics, the Klein four-group ( or just Klein group or Vierergruppe (), often symbolized by the letter V ) is the group Z < sub > 2 </ sub > × Z < sub > 2 </ sub >, the direct product of two copies of the cyclic group of order 2.

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

There is no prize awarded for mathematics, but see Abel Prize.