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

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

More generally, a binary relation between two sets A and B is a subset of.

The terms dyadic relation and 2-place relation are synonyms for binary relations.

The concept of function is defined as a special kind of binary relation.

A binary relation R is usually defined as an ordered triple ( X, Y, G ) where X and Y are arbitrary sets ( or classes ), and G is a subset of the Cartesian product X × Y.

Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

In mathematics, a directed set ( or a directed preorder or a filtered set ) is a nonempty set A together with a reflexive and transitive binary relation ≤ ( that is, a preorder ), with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > in X such that a

An equivalence relation is a binary relation satisfying three properties:

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function such that:

The binary abacus is used to explain how computers manipulate numbers.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

( A cancellative binary operation is enough.

A simple form of AM, often used for digital communications, is on-off keying: a type of amplitude-shift keying in which binary data is represented by the presence or absence of a carrier.

If there is a mixture of only two types of atoms, not counting impurities, such as a copper-nickel alloy, then it is called a binary alloy.

In this respect, all of the various forms of an alloy containing only two constituents, like iron and carbon, is called a binary system, while all of the alloy combinations possible with a ternary alloy, such as alloys of iron, carbon and chromium, is called a ternary system.

The observations show that the cloud is asymmetrical and matches the pattern of X-ray binaries ( binary star systems containing black holes or neutron stars ), mostly on one side of the galactic center.

In mathematics, the associative property is a property of some binary operations.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

* Absorbing element, in mathematics, an element that does not change when it is combined in a binary operation with some other element

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In computer science, an AVL tree is a self-balancing binary search tree, and it was the first such data structure to be invented.

In other words AVL tree is a binary search tree where the height of the left subtree differs from the height of the right subtree by at most 1 level, if it exceeds 1 level then rebalancing occurs.

Lookup in an AVL tree is performed exactly like in any unbalanced binary search tree.

As with all binary trees, a node's in-order successor is the left-most child of its right subtree, and a node's in-order predecessor is the right-most child of its left subtree.

For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O ( log ( n )), colloquially " in logarithmic time ".

The redesigned AIM for Mac is a full universal binary Cocoa API application that supports both Tiger and Leopard – Mac OS X 10. 4. 8 ( and above ) or Mac OS X 10. 5. 3 ( and above ).

However, Alpha Centauri is a bright binary star, whose unresolved components to the naked eye are both fainter than Arcturus.

Hipparcos also suggested that Arcturus is a binary star, with the companion about twenty times dimmer than the primary and orbiting close enough to be at the very limits of our current ability to make it out.

* RS CVn is the prototype of a special class of binary stars of chromospherically active and optically variable components.

Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen (" On the invariant properties of special binary forms, in particular the spherical harmonic functions ").

The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2:

In mathematics, an identity element ( or neutral element ) is a special type of element of a set with respect to a binary operation on that set.

MySQL can be built and installed manually from source code, but this can be tedious so it is more commonly installed from a binary package unless special customizations are required.

* Neutral element or identity element ( mathematics ), a special type of element of a set with respect to a binary operation on that set, such if applied to, or operated with any element of the set, causes no change to this last element

A red – black tree is a special type of binary tree, used in computer science to organize pieces of comparable data, such as text fragments or numbers.

Read-only operations on a red – black tree require no modification from those used for binary search trees, because every red – black tree is a special case of a simple binary search tree.

Finally, most of the special ( non-alphabetic ) characters can be used as what are called binary messages.

To improve performance for the special case of XML with embedded binary objects, the Message Transmission Optimization Mechanism was introduced.

Manchester encoding is a special case of binary phase-shift keying ( BPSK ), where the data controls the phase of a square wave carrier whose frequency is the data rate.

Since the definition also assures the existence of binary meets and joins, complete lattices do thus form a special class of bounded lattices.

In the special case of a binary partition,

We consider a special case of this theorem for a binary symmetric channel with an error probability p.

A special form of the LLN ( for a binary random variable ) was first proved by Jacob Bernoulli.

The case that the assigned value depends on previous one is so common that many imperative languages, most notably C and the majority of its descendants, augment the notion of assignment by defining special binary operators like for convenience so that the example becomes.

* arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers ( including signed zeros and subnormal numbers ), infinities, and special " not a number " values ( NaNs )

A special case, 0-1 integer linear programming, in which unknowns are binary, is one of Karp's 21 NP-complete problems.

The terminology seems to have been originated by Blum, although commitment schemes can be interchangeably called bit commitment schemes — sometimes reserved for the special case where the committed value is a binary bit.

Often the patch consists of bare binary data and a special program that replaces the previous version with the new version is provided.

From this, the generator matrix G can be obtained as ( noting that in the special case of this being a binary code ), or specifically:

PALs were programmed electrically using binary patterns ( as JEDEC ASCII / hexadecimal files ) and a special electronic programming system available from either the manufacturer or a third-party, such as DATA / IO.