Ordered pair in the context of Graph of a function of two variables

In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences".

Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning.

In mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.

An algebraic operation on a set ⁠${\displaystyle S}$⁠ may be defined more formally as a function that maps to ⁠${\displaystyle S}$⁠ the tuples of a given length of elements of ⁠${\displaystyle S}$⁠. The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain. Precisely, a binary relation over sets ${\displaystyle X}$ and ${\displaystyle Y}$ is a set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle x}$ is an element of ${\displaystyle X}$ and ${\displaystyle y}$ is an element of ${\displaystyle Y}$. It encodes the common concept of relation: an element ${\displaystyle x}$ is related to an element ${\displaystyle y}$, if and only if the pair ${\displaystyle (x,y)}$ belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set of prime numbers ${\displaystyle \mathbb {P} }$ and the set of integers ${\displaystyle \mathbb {Z} }$, in which each prime ${\displaystyle p}$ is related to each integer ${\displaystyle z}$ that is a multiple of ${\displaystyle p}$, but not to an integer that is not a multiple of ${\displaystyle p}$. In this relation, for instance, the prime number ${\displaystyle 2}$ is related to numbers such as ${\displaystyle -4}$, ${\displaystyle 0}$, ${\displaystyle 6}$, ${\displaystyle 10}$, but not to ${\displaystyle 1}$ or ${\displaystyle 9}$, just as the prime number ${\displaystyle 3}$ is related to ${\displaystyle 0}$, ${\displaystyle 6}$, and ${\displaystyle 9}$, but not to ${\displaystyle 4}$ or ${\displaystyle 13}$.

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.

More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.

In mathematics, the graph of a function ${\displaystyle f}$ is the set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle f(x)=y.}$ In the common case where ${\displaystyle x}$ and ${\displaystyle f(x)}$ are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve.The graphical representation of the graph of a function is also known as a plot.

In the case of functions of two variables – that is, functions whose domain consists of pairs ${\displaystyle (x,y)}$ –, the graph usually refers to the set of ordered triples ${\displaystyle (x,y,z)}$ where ${\displaystyle f(x,y)=z}$. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized as a surface plot.

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair ${\displaystyle P=(X,\leq )}$ consisting of a set ${\displaystyle X}$ (called the ground set of ${\displaystyle P}$) and a partial order ${\displaystyle \leq }$ on ${\displaystyle X}$. When the meaning is clear from context and there is no ambiguity about the partial order, the set ${\displaystyle X}$ itself is sometimes called a poset.

In mathematics, the abscissa (/æbˈsɪs.ə/; plural abscissae or abscissas) and the ordinate are respectively the first and second coordinate of a point in a Cartesian coordinate system:

Together they form an ordered pair which defines the location of a point in two-dimensional rectangular space.

In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable.

For example, the binary function ${\displaystyle f(x,y)=x^{2}+y^{2}}$ has two arguments, ${\displaystyle x}$ and ${\displaystyle y}$, in an ordered pair ${\displaystyle (x,y)}$. The hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function. A function that takes a single argument as input, such as ${\displaystyle f(x)=x^{2}}$, is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.

In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa.Set members may not be in relation "to a certain degree" – either they are in relation or they are not.

Formally, a relation R over a set X can be seen as a set of ordered pairs (x,y) of members of X.The relation R holds between x and y if (x,y) is a member of R.For example, the relation "is less than" on the natural numbers is an infinite set R_less of pairs of natural numbers that contains both (1,3) and (3,4), but neither (3,1) nor (4,4).The relation "is a nontrivial divisor of" on the set of one-digit natural numbers is sufficiently small to be shown here:R_dv = { (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) }; for example 2 is a nontrivial divisor of 8, but not vice versa, hence (2,8) ∈ R_dv, but (8,2) ∉ R_dv.

In semantics and pragmatics, a truth condition is the condition under which a sentence is true. For example, "It is snowing in Nebraska" is true precisely when it is snowing in Nebraska. Truth conditions of a sentence do not necessarily reflect current reality. They are merely the conditions under which the statement would be true.

More formally, a truth condition makes a sentence true for a given inductive definition of truth. Understood this way, truth conditions are theoretical entities. To illustrate with an example: suppose that, in a particular truth theory which is a theory of truth where truth is somehow made acceptable despite semantic terms as close as possible, the word "Nixon" refers to Richard M. Nixon, and "is alive" is associated with the set of currently living things. Then one way of representing the truth condition of "Nixon is alive" is as the ordered pair <Nixon, {x: x is alive}>. And we say that "Nixon is alive" is true if and only if the referent (or referent of) "Nixon" belongs to the set associated with "is alive", that is, if and only if Nixon is alive.

In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity.It is typically formulated as the product of a unit of measurement and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction.For example, a position vector in physical space may be expressed as three Cartesian coordinates with SI unit of meters.

In physics and engineering, particularly in mechanics, a physical vector may be endowed with additional structure compared to a geometrical vector.A bound vector is defined as the combination of an ordinary vector quantity and a point of application or point of action. Bound vector quantities are formulated as a directed line segment, with a definite initial point besides the magnitude and direction of the main vector.For example, a force on the Euclidean plane has two Cartesian components in SI unit of newtons (describing the magnitude and direction of the force) and an accompanying two-dimensional position vector in meters (describing the point of application of the force), for a total of four numbers on the plane (and six in space).A simpler example of a bound vector is the translation vector from an initial point to an end point; in this case, the bound vector is an ordered pair of points in the same position space, with all coordinates having the same quantity dimension and unit (length and meters).A sliding vector is the combination of an ordinary vector quantity and a line of application or line of action, over which the vector quantity can be translated (without rotations).A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector.

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. In terms of set-builder notation, that is$${\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.}$$

A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).