## domain

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## Domain

# Domain

The domain, in mathematics, of a relation is the set that contains all the first elements, x, from the ordered pairs (x, y) that make up the relation. In mathematics, a relation is defined as a set of ordered pairs (x, y) for which each y depends on x in a predetermined way. If x represents an element from the set X, and y represents an element from the set Y, the Cartesian product of X and Y is the set of all possible ordered pairs (x, y) that can be formed with an element of X being first. A relation between the sets X and Y is a subset of their Cartesian product, so the domain of the relation is a subset of the set X.

For example, suppose that X is the set of all men and Y is the set of all women. The Cartesian product of X and Y is the set of all ordered pairs having men first and women second. One of the many possible relations between these two sets is the set of all ordered pairs (x, y) such that x and y are married. The set of all married men is the domain of this relation, and is a subset of X. The set of all second elements from the ordered pairs of a relation is called the range of the relation, so the set of all married women is the range of this relation, and is a subset of Y. The variable associated with the domain of the relation is called the independent variable. The variable associated with the range of a relation is called the dependent variable.

Many important relations in science, engineering, business, and economics can be expressed as functions of real numbers. A function is a special type of relation in which none of the ordered pairs share the same first element. A real-valued function is a function between two sets X and Y, both of which correspond to the set of real numbers. The Cartesian product of these two sets is the familiar Cartesian coordinate system, with the set X associated with the x-axis and the set Y associated with the y-axis. The graph of a real-valued function consists of the set of points in the plane that are contained in the function, and thus represents a subset of the Cartesian plane. The x-axis, or some portion of it, corresponds to the domain of the function. Since, by definition, every set is a subset of itself, the domain of a function may correspond to the entire x-axis. In other cases the domain is limited to a portion of the x-axis, either explicitly or implicitly.

## Example 1

Let X and Y equal the set of real numbers. Let the function, *f*, be defined by the equation y = 3x^{2} + 2. Then, the variable x may range over the entire set of real numbers. That is, the domain of f is given by the set D = {xǀ-∞≤ x ≥∞}, read “D equals the set of all x such that negativeinfinity is less than or equal to x and x is less than or equal to infinity.”

## Example 2

Let X and Y equal the set of real numbers. Let the function f represent the location of a falling body during the second 5 seconds of descent. Then, letting t represent time (t), the location of the body, at any time between 5 and 10 seconds after descent begins, is given by f(t) = 1/2gt^{2}, where g is the acceleration due to gravity (g). In this example, the domain is explicitly limited to values of t between 5 and 10, that is, D = {tǀ 5≤ t≥ 5}.

## Example 3

Let X and Y equal the set of real numbers. Consider the function defined by y = πx^{2}, where y is the area of a circle and x is its radius. Since the radius of a circle cannot be negative, the domain, D, of this function is the set of all real numbers greater than or equal to zero, D = {xǀ x≥ 0}. In this example, the domain is limited implicitly by the physical circumstances.

## Example 4

Let X and Y equal the set of real numbers. Consider the function given by y = 1/x. The variable x can take on any real number value but zero, because division by zero is undefined. Hence the domain of this function is the set . Variations of this function exist, in which values of x other than zero make the denominator zero. The function defined by y = 1/2 - x is an example; x = 2 makes the denominator zero. In these examples the domain is again limited implicitly.

## Other Definitions in Science

In biology, the term domain is the highest level of scientifically classifying organisms. The classifications below domain are kingdom, phylum (or division), class, order, family, genus, species, and subspecies. In physics, domain is a region that has uniform magnetism throughout; that is, all its atoms are magnetically oriented in the same direction.

*See also* Cartesian coordinate plane.

## Resources

### BOOKS

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications*. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Grahm, Alan. *Teach Yourself Basic Mathematics*. Chicago, IL: McGraw-Hill Contemporary, 2001.

Jeffrey, Alan. *Mathematics for Engineers and Scientists*. Boca Raton, FL: Chapman & Hall/CRC, 2005.

Larson, Ron. *Calculus: An Applied Approach*. Boston, MA: Houghton Mifflin, 2003.

Lorenz, Falko. *Algebra*. New York: Springer, 2006.

Lyublinskava, Irina E. *Connecting Mathematics with Science: Experiments for Precalculus*. Emeryville, CA: Key Curriculum Press, 2003.

Setek, William M. *Fundamentals of Mathematics*. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J. R. Maddocks

## Domain

# Domain

The domain of a **relation** is the set that contains all the first elements, x, from the ordered pairs (x,y) that make up the relation. In **mathematics** , a relation is defined as a set of ordered pairs (x,y) for which each y depends on x in a predetermined way. If x represents an element from the set X, and y represents an element from the set Y, the Cartesian product of X and Y is the set of all possible ordered pairs (x,y) that can be formed with an element of X being first. A relation between the sets X and Y is a subset of their Cartesian product, so the domain of the relation is a subset of the set X. For example, suppose that X is the set of all men and Y is the set of all women. The Cartesian product of X and Y is the set of all ordered pairs having a man first and women second. One of the many possible relations between these two sets is the set of all ordered pairs (x,y) such that x and y are married. The set of all married men is the domain of this relation, and is a subset of X. The set of all second elements from the ordered pairs of a relation is called the range of the relation, so the set of all married women is the range of this relation, and is a subset of Y. The **variable** associated with the domain of the relation is called the independent variable. The variable associated with the range of a relation is called the dependent variable.

Many important relations in science, **engineering** , business and economics can be expressed as functions of **real numbers** . A **function** is a special type of relation in which none of the ordered pairs share the same first element. A real-valued function is a function between two sets X and Y, both of which correspond to the set of real numbers. The Cartesian product of these two sets is the familiar Cartesian coordinate system, with the set X associated with the x-axis and the set Y associated with the y-axis. The graph of a real-valued function consists of the set of points in the **plane** that are contained in the function, and thus represents a subset of the Cartesian plane. The x-axis, or some portion of it, corresponds to the domain of the function. Since, by definition, every set is a subset of itself, the domain of a function may correspond to the entire x-axis. In other cases the domain is limited to a portion of the x-axis, either explicitly or implicitly.

Example 1. Let X and Y equal the set of real numbers. Let the function, f, be defined by the equation y= 3x2 + 2. Then the variable x may range over the entire set of real numbers. That is, the domain of f is given by the set D = {x| - ∞ ≤ x ≥ ∞}, read "D equals the set of all x such that **negative infinity** is less than or equal to x and x is less than or equal to infinity."

Example 2. Let X and Y equal the set of real numbers. Let the function f represent the location of a falling body during the second 5 seconds of descent. Then, letting t represent **time** , the location of the body, at any time between 5 and 10 seconds after descent begins, is given by f(t) = 1⁄2gt2. In this example, the domain is explicitly limited to values of t between 5 and 10, that is, D = {t| 5 ≤ t ≥ 5}.

Example 3. Let X and Y equal the set of real numbers. Consider the function defined by y = PIx2, where y is the area of a **circle** and x is its radius. Since the radius of a circle cannot be negative, the domain, D, of this function is the set of all real numbers greater than or equal to **zero** , D = {x| x ≥ 0}. In this example, the domain is limited implicitly by the physical circumstances.

Example 4. Let X and Y equal the set of real numbers. Consider the function given by y = 1/x. The variable x can take on any real number value but zero, because **division** by zero is undefined. Hence the domain of this function is the set D = {x| x NSIME 0}. Variations of this function exist, in which values of x other than zero make the denominator zero. The function defined by y = 1⁄2-x is an example; x=2 makes the denominator zero. In these examples the domain is again limited implicitly.

See also Cartesian coordinate plane.

## Resources

### books

Allen, G.D., C. Chui, and B. Perry. *Elements of Calculus.* 2nd ed. Pacific Grove, CA.: Brooks/Cole Publishing Co., 1989.

Bittinger, Marvin L., and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Grahm, Alan. *Teach Yourself Basic Mathematics.* Chicago: McGraw-Hill Contemporary, 2001.

Swokowski, Earl W. *Pre Calculus, Functions, and Graphs, 6th ed.* Boston: PWS-KENT Publishing Co., 1990.

J. R. Maddocks

## domain

**domain** **1.** In general, a sphere of control, influence, or concern.

**2.** See function, relation, category. See also range.

**3.** of a network. Part of a larger network. A domain is usually defined in terms of some property, such as that part of the network that is under the jurisdiction of a single management body (a *management domain*), or where all the network addresses are assigned by a single controlling authority (a *naming domain*). See also domain name server.

**4.** In the relational model, a set of possible values from which the actual values in any column of a table (relation) must be drawn.

**5.** In denotational semantics, a structured set of mathematical entities in which meanings for programming constructs can be found. The idea first arose in the work of Dana Scott, who with Christopher Strachey pioneered this mathematical approach to programming language semantics. The approach focuses on fixed-point theorems. Scott required domains to be complete lattices, but this has been simplified through a great deal of mathematical research. There are now many kinds of domains, but a commonly used one is the *Scott–Ershov domain*, which is a consistently complete algebraic cpo (complete partial ordering). For such mathematical structures a fine theory of constructing new domains from old and solving fixed-point equations has been developed. The *domain theory* has many applications in finding semantics for programming and specification languages, and approximating data types. Mathematically the theory is closely linked to topology and algebra.

**6.** See protection domain.

## domain

do·main
/ dōˈmān/
•
n.
an area of territory owned or controlled by a ruler or government:
*the southwestern French domains of the Plantagenets.*
∎
an estate or territory held in legal possession by a person or persons.
∎
a specified sphere of activity or knowledge:
*the expanding domain of psychology* |
fig.
*visual communication is the domain of the graphic designer.*
∎
Physics
a discrete region of magnetism in ferromagnetic material.
∎
Comput.
a distinct subset of the Internet with addresses sharing a common suffix, such as the part in a particular country or used by a particular group of users.
∎
Math.
the set of possible values of the independent variable or variables of a function.
DERIVATIVES:
do·ma·ni·al
/ -nēəl/ adj.

## domain

**domain**
•**abstain**, appertain, arcane, arraign, ascertain, attain, Bahrain, bane, blain, brain, Braine, Cain, Caine, campaign, cane, chain, champagne, champaign, Champlain, Charmaine, chicane, chow mein, cocaine, Coleraine, Coltrane, complain, constrain, contain, crane, Dane, deign, demesne, demi-mondaine, detain, disdain, domain, domaine, drain, Duane, Dwane, Elaine, entertain, entrain, explain, fain, fane, feign, gain, Germaine, germane, grain, humane, Hussein, inane, Jain, Jane, Jermaine, Kane, La Fontaine, lain, lane, legerdemain, Lorraine, main, Maine, maintain, mane, mise en scène, Montaigne, moraine, mundane, obtain, ordain, pain, Paine, pane, pertain, plain, plane, Port-of-Spain, profane, rain, Raine, refrain, reign, rein, retain, romaine, sane, Seine, Shane, Sinn Fein, skein, slain, Spain, Spillane, sprain, stain, strain, sustain, swain, terrain, thane, train, twain, Ujjain, Ukraine, underlain, urbane, vain, vane, vein, Verlaine, vicereine, wain, wane, Wayne
•watch chain • mondaine • Haldane
•ultramundane • Cellophane
•novocaine • sugar cane • marocain

## domain

**domain** **1.** (in biochemistry) A functional unit of the tertiary structure of a protein. It consists of chains of amino acids folded into alpha helices and beta sheets to form a globular structure. Different domains are linked together by relatively straight sections of polypeptide chain to form the protein molecule. Domains allow a degree of movement in the protein structure. See also finger domain.

**2.** (in taxonomy) In some classification systems, the highest taxonomic category, consisting of one or more kingdoms. Some authors divide living organisms into three domains: Archaea (archaebacteria), Bacteria (eubacteria; see bacteria), and Eukarya (eukaryotic organisms).

## domain

**domain** In taxonomy, some experts recognize the domain as a higher category than kingdom. In this scheme, the two subkingdoms of Prokaryote (Archaebacteria and Eubacteria) constitute two domains, called Archaea and Bacteria, while all other living organisms are in a third domain, eukaryotes. See also phylogenetics; plant classification

## Domain

# DOMAIN

*The complete and absolute ownership of land. Also the real estate so owned. The inherent sovereign power claimed by the legislature of a state, of controlling private property for public uses, is termed the* right of *eminent domain.*

*National domain is sometimes applied to the aggregate of the property owned directly by a nation. Public domain embraces all lands, the title to which is in the United States, including land occupied for the purposes of federal buildings, arsenals, dock-yards, and so on, and land of an agricultural or mineral character not yet granted to private owners.*

*Sphere of influence. Range of control or rule; realm.*

## domain

**domain** In mathematics, a set of values that can be assigned to the independent variable in a function or relation; the set of values of the dependent variable is called the *range*. For example, let the function be y = x^{2}, with *x* restricted to 0, 1, 2, 3 and −3. Then *y* takes the values 0, 1, 4, 9 and 9 respectively. The domain is {0, 1, 2, 3, −3} and the range is {0, 1, 4, 9}.

## domain

**domain** The primary division of living systems. There are three domains: Archaea, Eubacteria and Eucaryota.