Some adaptations of the Latin alphabet are augmented with ligatures, such as æ in Old English and Icelandic and Ȣ in Algonquian ; by borrowings from other alphabets, such as the thorn þ in Old English and Icelandic, which came from the Futhark runes ; and by modifying existing letters, such as the eth ð of Old English and Icelandic, which is a modified d. Other alphabets only use a subset of the Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x, y and w only in foreign words.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.

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.

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** 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.

** Every infinite game in which is a Borel subset of Baire space is determined.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

Some say that literary criticism is a subset of literary theory.

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:

In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable.

* A nonempty compact subset of the real numbers has a greatest element and a least element.

* The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal ( and must be contained in every other cofinal subset ).

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

In mathematics, the infimum ( plural infima ) of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound ( also abbreviated as glb or GLB ) is also commonly used.

In analysis the infimum or greatest lower bound of a subset S of real numbers is denoted by inf ( S ) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists ( because S is not bounded below ), then we define inf ( S ) = −∞.

If S contains an upper bound then that upper bound is unique and is called the greatest element of S. The greatest element of S ( if it exists ) is also the least upper bound of S. A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds.

A partially ordered set ( L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound ( the infimum, also called the meet ) and a least upper bound ( the supremum, also called the join ) in ( L, ≤).

A bounded poset P ( that is, by itself, not as subset ) is one that has a least element and a greatest element.

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

* Infimum, the greatest lower bound of a subset of a partially ordered set

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set ( poset ) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually.

Formally, given a partially ordered set ( P, ≤), then an element g of a subset S of P is the greatest element of S if

Hence, the greatest element of S is an upper bound of S that is contained within this subset.

However, there is here a similar effect to “ multi-tasking ”, as workers shift effort from that subset of tasks which they consider useful and constructive, to that subset which they think gives the greatest appearance of being useful and constructive, and more generally to try to curry personal favour with supervisors.

Dually, a meet-semilattice ( or lower semilattice ) is a partially ordered set which has a meet ( or greatest lower bound ) for any nonempty finite subset.

For a partially ordered set with greatest element, a subset is cofinal if and only if it contains that greatest element.

* Infimum ( also greatest lower bound ), a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S

It is fair to say that many ( but not all ) of these criticisms can only be directed towards a subset of the neoclassical models ( for example, there are many neoclassical models where unregulated markets fail to achieve Pareto-optimality and there has recently been an increased interest in modeling non-rational decision making ).

In contrast, the ideal sanity test exercises the smallest subset of application functions needed to determine whether the application logic is generally functional and correct ( for example, an interest rate calculation for a financial application ).

Not every subset of the sample space Ω must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be “ measured ”.

Stockholders or shareholders are considered by some to be a subset of stakeholders, which may include anyone who has a direct or indirect interest in the business entity.

" The interest … is not in investigating a mathematically definable system which has some relation to language, as being a generalization or a subset of it, but in formulating as a mathematical system all the properties and relations necessary and sufficient for the whole of natural language.

Organizations which are not technical may also have Special Interest Groups which are normally focused on a mutual interest or shared characteristic of a subset of members of the organization.

In many applications an analysis may start with a given collection of random variables, then first extend the set by defining new ones ( such as the sum of the original random variables ) and finally reduce the number by placing interest in the marginal distribution of a subset ( such as the sum ).

Based on the original definition of Weiser, informally, a static program slice S consists of all statements in program P that may affect the value of variable v at some point p. The slice is defined for a slicing criterion C =( x, V ), where x is a statement in program P and V is a subset of variables in P. A static slice includes all the statements that affect variable v for a set of all possible inputs at the point of interest ( i. e., at the statement x ).

Most software allows saving only a subset of recorded frames, minimizing file size issues by eliminating useless frames before or after the sequence of interest.

Some software allows viewing the issues in real time, by displaying only a subset of recorded frames, minimizing file size and watch time issues by eliminating useless frames before or after the sequence of interest.

In a partition by allotment, which is not available in all jurisdictions, the court awards full ownership of the land to a single owner or subset of owners, and orders them to pay the person or persons divested of ownership for the interest awarded.

A domain model may also include a number of conceptual views, where each view is pertinent to a particular subject area of the domain or to a particular subset of the domain model which is of interest to a stakeholder of the domain model.