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In mathematics, the domain of a function is the set of all input values to the function. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
Partial plot of a function f. ...
Domain of a function
Given a function f:X→Y, the set X of input values is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...
In mathematics, the range of a function is the set of all output values produced by that function. ...
A well-defined function must map every element of the domain to an element of its codomain. For example, the function f defined by
- f(x) = 1/x
has no value for f(0). Thus, the set R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R{0}, or the "gap" is plugged by specifically defining f(0). If we extend the definition of f to In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
- f(x) = 1/x, for x ≠ 0
- f(0) = 0,
then f is defined for all real numbers and we can choose its domain to be R.
Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |S : S → B. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...
Some well-known domains are as follows (note that each successive domain includes those above it):
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
Please refer to Real vs. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined. Some (particularly category theorists), however, consider the domain of a partial function f:X→Y to be X, irrespective of whether f(x) exists for all x in X. This article needs to be cleaned up to conform to a higher standard of quality. ...
Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Category theory In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to be abandoned, or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ...
Complex analysis In complex analysis, a domain is an open connected subset of the complex numbers. Complex analysis is the branch of mathematics investigating functions of complex numbers. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ...
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