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Variable (mathematics)

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Variable_(computer_science)" title="Variable (computer science)">Variable (computer science)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a variable (from <a href="Latin_language" class="mw-redirect" title="Latin language">Latin</a> <a href="https://en.wiktionary.org/wiki/variabilis#Latin" class="extiw external" title="wikt:variabilis">variabilis</a> 'changeable') is a <a href="Mathematical_symbol" class="mw-redirect" title="Mathematical symbol">symbol</a>, typically a letter, that refers to an unspecified <a href="Mathematical_object" title="Mathematical object">mathematical object</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a <a href="Set_(mathematics)" title="Set (mathematics)">set</a>, such as the set of <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>.

The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables p and q and require that the value of the square of p is twice the square of q, which in algebraic notation can be written p2 = 2 q2. A definitive proof that this relationship is impossible to satisfy when p and q are restricted to integer numbers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.

Originally, the term variable was used primarily for the <a href="Argument_of_a_function" title="Argument of a function">argument of a function</a>, in which case its value could be thought of as varying within the <a href="Domain_of_a_function" title="Domain of a function">domain of the function</a>. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as the symbol y in the equation y = f(x), where x is the argument and f denotes the function itself.

A variable may represent an unspecified number that remains fixed during the resolution of a problem; in which case, it is often called a <a href="Parameter" title="Parameter">parameter</a>. A variable may denote an unknown number that has to be determined; in which case, it is called an <a href="Unknown_(mathematics)" class="mw-redirect" title="Unknown (mathematics)">unknown</a>; for example, in the <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> ax2 + bx + c = 0, the variables a, b, c are parameters, and x is the unknown.

Sometimes the same symbol can be used to denote both a variable and a <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a>, that is a well defined mathematical object. For example, the <a href="Greek_letter" class="mw-redirect" title="Greek letter">Greek letter</a> π generally represents the number <a href="Pi" title="Pi">π</a>, but has also been used to denote a <a href="Projection_(mathematics)" title="Projection (mathematics)">projection</a>. Similarly, the letter e often denotes <a href="E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a>, but has been used to denote an unassigned <a href="Coefficient" title="Coefficient">coefficient</a> for <a href="Quartic_function" title="Quartic function">quartic function</a> and higher <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree polynomials</a>. Even the symbol 1 has been used to denote an <a href="Identity_element" title="Identity element">identity element</a> of an arbitrary <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant.<a href="#cite_note-80000-2:2019-4">[4]</a>

Variables are often used for representing <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, <a href="Function_(mathematics)" title="Function (mathematics)">functions</a>, their arguments, <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> and their <a href="Element_(mathematics)" title="Element (mathematics)">elements</a>, <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>, <a href="Space_(mathematics)" title="Space (mathematics)">spaces</a>, etc.<a href="#cite_note-5">[5]</a>

In <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, a variable is a symbol that either represents an unspecified constant of the theory, or is being <a href="Quantifier_(logic)" title="Quantifier (logic)">quantified</a> over.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="History_of_algebra" title="History of algebra">History of algebra</a> and <a href="History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a>.</div>

<div class="mw-heading mw-heading3"><h3 id="Early_history">Early history</h3></div>

The earliest uses of an "unknown quantity" date back to at least the <a href="Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egyptians</a> with the <a href="Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a> (c. 1500 BC) which described problems with unknowns rhetorically, called the "Aha problems". The "Aha problems" involve finding unknown quantities (referred to as aha, "stack") if the sum of the quantity and part(s) of it are given (The <a href="Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> also contains four of these types of problems). For example, problem 19 asks one to calculate a quantity taken <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>1+1⁄2 times and added to 4 to make 10.<a href="#cite_note-Clagett-9">[9]</a> In modern mathematical notation: <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠3/2⁠x + 4 = 10. Around the same time in Mesopotamia, <a href="Babylonian_mathematics" title="Babylonian mathematics">mathematics of the Old Babylonian period</a> (c. 2000 BC – 1500 BC) was more advanced, also studying quadratic and <a href="Cubic_equation" title="Cubic equation">cubic equations</a>.<a href="#cite_note-:0-10">[10]</a>

In works of <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">ancient greece</a> such as <a href="Euclid's_Elements" title="Euclid's Elements">Euclid's Elements</a> (c. 300 BC), mathematics was described <a href="Geometry" title="Geometry">geometrically</a>. For example, The Elements, proposition 1 of Book II, <a href="Euclid" title="Euclid">Euclid</a> includes the proposition:

"If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments."

This corresponds to the algebraic identity a(b + c) = ab + ac (<a href="Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>), but is described entirely geometrically. Euclid, and other greek geometers, also used single letters refer to geometric points and shapes. This kind of algebra is now sometimes called <a href="Greek_geometric_algebra" class="mw-redirect" title="Greek geometric algebra">Greek geometric algebra</a>.<a href="#cite_note-:0-10">[10]</a>

<a href="Diophantus" title="Diophantus">Diophantus</a> of <a href="History_of_Alexandria" title="History of Alexandria">Alexandria</a>,<a href="#cite_note-11">[11]</a> pioneered a form of <a href="Syncopated_algebra" class="mw-redirect" title="Syncopated algebra">syncopated algebra</a> in his <a href="Arithmetica" class="mw-redirect" title="Arithmetica">Arithmetica</a> (c. 200 AD), which introduced symbolic manipulation of expressions with unknowns and powers, but without modern symbols for <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a> (such as <a href="Equality_(mathematics)" title="Equality (mathematics)">equality</a> or <a href="Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a>) or <a href="Exponentiation" title="Exponentiation">exponents</a>.<a href="#cite_note-Boyer2-12">[12]</a> An unknown number was called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy">.<a href="#cite_note-13">[13]</a> The square of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{v}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta ^{v}}</annotation>

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</math><img src="./bc4c7e133f61d18dff15b36ff814bbb2b92648c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.965ex; height:2.343ex;" alt="{\displaystyle \Delta ^{v}}" loading="lazy">; the cube was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{v}}">

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<annotation encoding="application/x-tex">{\displaystyle K^{v}}</annotation>

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</math><img src="./fce4d49f974be73816365cf9b1fc7900c90c12b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.124ex; height:2.343ex;" alt="{\displaystyle K^{v}}" loading="lazy">; the fourth power was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{v}\Delta }">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta ^{v}\Delta }</annotation>

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</math><img src="./7a4e3907f0ccaa0acce4d02357b28c7ab5d28162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.901ex; height:2.343ex;" alt="{\displaystyle \Delta ^{v}\Delta }" loading="lazy">; and the fifth power was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta K^{v}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta K^{v}}</annotation>

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</math><img src="./c10ff4e71e861767c5caa61ee46f3865081e3266.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.059ex; height:2.343ex;" alt="{\displaystyle \Delta K^{v}}" loading="lazy">.<a href="#cite_note-14">[14]</a> So for example, what would be written in modern notation as:

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{3}-2x^{2}+10x-1,}</annotation>

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would be written in Diophantus's syncopated notation as:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}</annotation>

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</math><img src="./6a6f34b992f64ad82a6963fea1a3f486c0d7a67e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.039ex; height:3.343ex;" alt="{\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}" loading="lazy"></dd></dl>

In the 7th century BC, <a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a> used different colours to represent the unknowns in algebraic equations in the <a href="Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a>. One section of this book is called "Equations of Several Colours".<a href="#cite_note-15">[15]</a> Greek and other ancient mathematical advances, were often trapped in long periods of stagnation, and so there were few revolutions in notation, but this began to change by the <a href="Early_modern_period" title="Early modern period">early modern period</a>.

<div class="mw-heading mw-heading3"><h3 id="Early_modern_period">Early modern period</h3></div>

At the end of the 16th century, <a href="Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns.<a href="#cite_note-16">[16]</a>

In 1637, <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c".<a href="#cite_note-17">[17]</a> Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887 <a href="Scientific_American" title="Scientific American">Scientific American</a> article.<a href="#cite_note-18">[18]</a>

Starting in the 1660s, <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> independently developed the <a href="Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a>, which essentially consists of studying how an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> variation of a time-varying quantity, called a <a href="Fluent_(mathematics)" title="Fluent (mathematics)">Fluent</a>, induces a corresponding variation of another quantity which is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of the first variable. Almost a century later, <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> fixed the terminology of infinitesimal calculus, and introduced the notation y = f(x) for a function f, its variable x and its value y. Until the end of the 19th century, the word variable referred almost exclusively to the <a href="Argument_of_a_function" title="Argument of a function">arguments</a> and the <a href="Value_(mathematics)" title="Value (mathematics)">values</a> of functions.

In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere <a href="Differentiable_function" title="Differentiable function">differentiable</a> <a href="Continuous_function" title="Continuous function">continuous function</a>. To solve this problem, <a href="Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> introduced a new formalism consisting of replacing the intuitive notion of <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> by a formal definition. The older notion of limit was "when the variable x varies and tends toward a, then f(x) tends toward L", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall \epsilon >0)(\exists \eta >0)(\forall x)\;|x-a|<\eta }">

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<annotation encoding="application/x-tex">{\displaystyle (\forall \epsilon >0)(\exists \eta >0)(\forall x)\;|x-a|<\eta }</annotation>

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</math><img src="./f7f9bb83abf3ad93c4d5bc10bbec43951b0e573c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.877ex; height:2.843ex;" alt="{\displaystyle (\forall \epsilon >0)(\exists \eta >0)(\forall x)\;|x-a|<\eta }" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\Rightarrow |L-f(x)|<\epsilon ,}">

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<annotation encoding="application/x-tex">{\displaystyle \;\Rightarrow |L-f(x)|<\epsilon ,}</annotation>

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</math><img src="./63995e22c1840452bb1a152b701e55a34ef873de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.438ex; height:2.843ex;" alt="{\displaystyle \;\Rightarrow |L-f(x)|<\epsilon ,}" loading="lazy"></dd></dl>

in which none of the five variables is considered as varying.

This static formulation led to the modern notion of variable, which is simply a symbol representing a <a href="Mathematical_object" title="Mathematical object">mathematical object</a> that either is unknown, or may be replaced by any element of a given <a href="Set_(mathematics)" title="Set (mathematics)">set</a> (e.g., the set of <a href="Real_number" title="Real number">real numbers</a>).

<div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2></div>

Variables are generally denoted by a single letter, most often from the <a href="Latin_alphabet" title="Latin alphabet">Latin alphabet</a> and less often from the <a href="Greek_alphabet" title="Greek alphabet">Greek</a>, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in x2), another variable (xi), a word or abbreviation of a word as a label (xtotal) or a <a href="Mathematical_expression" class="mw-redirect" title="Mathematical expression">mathematical expression</a> (x2i+1). Under the influence of <a href="Variable_(computer_science)" title="Variable (computer science)">computer science</a>, some variable names in pure mathematics consist of several letters and digits. Following <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (1596–1650), letters at the beginning of the alphabet such as a, b, c are commonly used for known values and parameters, and letters at the end of the alphabet such as x, y, z are commonly used for unknowns and variables of functions.<a href="#cite_note-E004-19">[19]</a> In printed mathematics, the norm is to set variables and constants in an italic typeface.<a href="#cite_note-20">[20]</a>

For example, a general <a href="Quadratic_function" title="Quadratic function">quadratic function</a> is conventionally written as ax2 + bx + c, where a, b and c are parameters (also called <a href="Constant_(mathematics)" title="Constant (mathematics)">constants</a>, because they are <a href="Constant_function" title="Constant function">constant functions</a>), while x is the variable of the function. A more explicit way to denote this function is x ↦ ax2 + bx + c, which clarifies the function-argument status of x and the constant status of a, b and c. Since c occurs in a term that is a constant function of x, it is called the <a href="Constant_term" title="Constant term">constant term</a>.<a href="#cite_note-21">[21]</a>

Specific branches and applications of mathematics have specific <a href="Naming_conventions" class="mw-redirect" title="Naming conventions">naming conventions</a> for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D <a href="Coordinate_space" class="mw-redirect" title="Coordinate space">coordinate space</a> are conventionally called x, y, and z. In physics, the names of variables are largely determined by the <a href="Physical_quantity" title="Physical quantity">physical quantity</a> they describe, but various naming conventions exist. A convention often followed in <a href="Probability" title="Probability">probability</a> and <a href="Statistics" title="Statistics">statistics</a> is to use X, Y, Z for the names of <a href="Random_variable" title="Random variable">random variables</a>, keeping x, y, z for variables representing corresponding better-defined values.

<div class="mw-heading mw-heading3"><h3 id="Conventional_variable_names">Conventional variable names</h3></div>

<ul><li>a, b, c, d (sometimes extended to e, f) for parameters or <a href="Coefficient" title="Coefficient">coefficients</a></li>

<li>a0, a1, a2, ... for situations where distinct letters are inconvenient</li>

<li>ai or ui for the ith term of a <a href="Sequence" title="Sequence">sequence</a> or the ith coefficient of a <a href="Series_(mathematics)" title="Series (mathematics)">series</a></li>

<li>f, g, h for <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> (as in f(x))</li>

<li>i, j, k (sometimes l or h) for varying <a href="Integers" class="mw-redirect" title="Integers">integers</a> or indices in an <a href="Indexed_family" title="Indexed family">indexed family</a>, or <a href="Unit_vectors" class="mw-redirect" title="Unit vectors">unit vectors</a></li>

<li>l and w for the length and width of a figure</li>

<li>l also for a line, or in number theory for a prime number not equal to p</li>

<li>n (with m as a second choice) for a fixed integer, such as a count of objects or the <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree of a polynomial</a></li>

<li>p for a <a href="Prime_number" title="Prime number">prime number</a> or a <a href="Probability" title="Probability">probability</a></li>

<li>q for a <a href="Prime_power" title="Prime power">prime power</a> or a <a href="Quotient" title="Quotient">quotient</a></li>

<li>r for a <a href="Radius" title="Radius">radius</a>, a <a href="Remainder" title="Remainder">remainder</a> or a <a href="Correlation_coefficient" title="Correlation coefficient">correlation coefficient</a></li>

<li>x, y, z for the three <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of a point in <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> or the corresponding <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">axes</a></li>

<li>z for a <a href="Complex_number" title="Complex number">complex number</a>, or in statistics a <a href="Normal_distribution" title="Normal distribution">normal random variable</a></li>

<li>α, β, γ, θ, φ for <a href="Angle_measure" class="mw-redirect" title="Angle measure">angle measures</a></li>

<li>ε (with δ as a second choice) for an arbitrarily small positive number</li>

<li>λ for an <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a></li>

<li>Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the <a href="Standard_deviation" title="Standard deviation">standard deviation</a><a href="#cite_note-22">[22]</a></li>

<div class="mw-heading mw-heading2"><h2 id="Specific_kinds_of_variables">Specific kinds of variables</h2></div>

It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general <a href="Cubic_equation" title="Cubic equation">cubic equation</a>

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<annotation encoding="application/x-tex">{\displaystyle ax^{3}+bx^{2}+cx+d=0,}</annotation>

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</math><img src="./4c864462f7d6bda7015dffbbfeb51127294147e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.977ex; height:3.009ex;" alt="{\displaystyle ax^{3}+bx^{2}+cx+d=0,}" loading="lazy"></dd></dl>

is interpreted as having five variables: four, a, b, c, d, which are taken to be given numbers and the fifth variable, x, is understood to be an unknown number. To distinguish them, the variable x is called an unknown, and the other variables are called parameters or <a href="Coefficient" title="Coefficient">coefficients</a>, or sometimes constants, although this last terminology is incorrect for an equation, and should be reserved for the <a href="Function_(mathematics)" title="Function (mathematics)">function</a> defined by the left-hand side of this equation.

In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "<a href="Function_of_a_real_variable" title="Function of a real variable">function of a real variable</a>", "x is the variable of the function f : x ↦ f(x)", "f is a function of the variable x" (meaning that the argument of the function is referred to by the variable x).

In the same context, variables that are independent of x define <a href="Constant_function" title="Constant function">constant functions</a> and are therefore called constant. For example, a <a href="Constant_of_integration" title="Constant of integration">constant of integration</a> is an arbitrary constant function that is added to a particular <a href="Antiderivative" title="Antiderivative">antiderivative</a> to obtain the other antiderivatives. Because of the strong relationship between <a href="Polynomial" title="Polynomial">polynomials</a> and <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a>, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

Other specific names for variables are:

<ul><li>An unknown is a variable in an <a href="Equation" title="Equation">equation</a> which has to be solved for.</li>

<li>An <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminate</a> is a symbol, commonly called variable, that appears in a <a href="Polynomial" title="Polynomial">polynomial</a> or a <a href="Formal_power_series" title="Formal power series">formal power series</a>. Formally speaking, an indeterminate is not a variable, but a <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a> in the <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> or the ring of <a href="Formal_power_series" title="Formal power series">formal power series</a>. However, because of the strong relationship between polynomials or <a href="Power_series" title="Power series">power series</a> and the <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> that they define, many authors consider indeterminates as a special kind of variables.</li>

<li>A <a href="Parameter" title="Parameter">parameter</a> is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in <a href="Mechanics" title="Mechanics">mechanics</a> the mass and the size of a solid body are parameters for the study of its movement. In <a href="Computer_science" title="Computer science">computer science</a>, parameter has a different meaning and denotes an argument of a function.</li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li>

<li>A <a href="Random_variable" title="Random variable">random variable</a> is a kind of variable that is used in <a href="Probability_theory" title="Probability theory">probability theory</a> and its applications.</li></ul>

All these denominations of variables are of <a href="Semantics" title="Semantics">semantic</a> nature, and the way of computing with them (<a href="Syntax_(logic)" title="Syntax (logic)">syntax</a>) is the same for all.

<div class="mw-heading mw-heading3"><h3 id="Dependent_and_independent_variables">Dependent and independent variables</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></div>

In <a href="Calculus" title="Calculus">calculus</a> and its application to <a href="Physics" title="Physics">physics</a> and other sciences, it is rather common to consider a variable, say y, whose possible values depend on the value of another variable, say x. In mathematical terms, the dependent variable y represents the value of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of x. To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y. For example, the state of a physical system depends on measurable quantities such as the <a href="Pressure" title="Pressure">pressure</a>, the <a href="Temperature" title="Temperature">temperature</a>, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.

Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent.<a href="#cite_note-23">[23]</a>

The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation f(x, y, z), the three variables may be all independent and the notation represents a function of three variables. On the other hand, if y and z depend on x (are dependent variables) then the notation represents a function of the single independent variable x.<a href="#cite_note-24">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

If one defines a function f from the <a href="Real_number" title="Real number">real numbers</a> to the real numbers by

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<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+\sin(x+4)}</annotation>

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</math><img src="./963531657fac2761bdf23fc4becd7d55784505cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.738ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+\sin(x+4)}" loading="lazy"></dd></dl>

then x is a variable standing for the <a href="Argument_of_a_function" title="Argument of a function">argument</a> of the function being defined, which can be any real number.

In the identity

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}i={\frac {n^{2}+n}{2}}}</annotation>

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</math><img src="./171c94205af7d6ea67fa0c2ae8eaa15d67c74332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.163ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}i={\frac {n^{2}+n}{2}}}" loading="lazy"></dd></dl>

the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).

In the theory of <a href="Polynomials" class="mw-redirect" title="Polynomials">polynomials</a>, a polynomial of degree 2 is generally denoted as ax2 + bx + c, where a, b and c are called <a href="Coefficient" title="Coefficient">coefficients</a> (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

<div class="mw-heading mw-heading4"><h4 id="Example:_the_ideal_gas_law">Example: the ideal gas law</h4></div>

Consider the equation describing the ideal gas law,

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<annotation encoding="application/x-tex">{\displaystyle PV=Nk_{\text{B}}T.}</annotation>

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This equation would generally be interpreted to have four variables, and one constant. The constant is kB, the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>. One of the variables, N, the number of particles, is a positive integer (and therefore a discrete variable), while the other three, P, V and T, for pressure, volume and temperature, are continuous variables.

One could rearrange this equation to obtain P as a function of the other variables,

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<annotation encoding="application/x-tex">{\displaystyle P(V,N,T)={\frac {Nk_{\text{B}}T}{V}}.}</annotation>

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Then P, as a function of the other variables, is the dependent variable, while its arguments, V, N and T, are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here P is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:\mathbb {R} _{>0}\times \mathbb {N} \times \mathbb {R} _{>0}\rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle P:\mathbb {R} _{>0}\times \mathbb {N} \times \mathbb {R} _{>0}\rightarrow \mathbb {R} }</annotation>

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</math><img src="./1dc17207d6dcc1ff0f1e21df2787ce8f8d6ecac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.355ex; height:2.509ex;" alt="{\displaystyle P:\mathbb {R} _{>0}\times \mathbb {N} \times \mathbb {R} _{>0}\rightarrow \mathbb {R} }" loading="lazy">.

However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say T. This gives a function

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<annotation encoding="application/x-tex">{\displaystyle P(T)={\frac {Nk_{\text{B}}T}{V}},}</annotation>

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where now N and V are also regarded as constants. Mathematically, this constitutes a <a href="Partial_application" title="Partial application">partial application</a> of the earlier function P.

This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard kB as a variable to obtain a function

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<annotation encoding="application/x-tex">{\displaystyle P(V,N,T,k_{\text{B}})={\frac {Nk_{\text{B}}T}{V}}.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Moduli_spaces">Moduli spaces</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Moduli_spaces" class="mw-redirect" title="Moduli spaces">moduli spaces</a></div>

Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a <a href="Parabola" title="Parabola">parabola</a>,

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<annotation encoding="application/x-tex">{\displaystyle y=ax^{2}+bx+c,}</annotation>

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where a, b, c, x and y are all considered to be real. The set of points (x, y) in the 2D plane satisfying this equation trace out the graph of a parabola. Here, a, b and c are regarded as constants, which specify the parabola, while x and y are variables.

Then instead regarding a, b and c as variables, we observe that each set of 3-tuples (a, b, c) corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas.

<ul><li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Observable_variable" class="mw-redirect" title="Observable variable">Observable variable</a></li>

<li><a href="Physical_constant" title="Physical constant">Physical constant</a></li>

<li><a href="Propositional_variable" title="Propositional variable">Propositional variable</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFLandin1989" class="citation book cs1">Landin, Joseph (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00land/page/2/mode/2up?q=variable">An Introduction to Algebraic Structures</a>. New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. p. 204. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-65940-2</bdi>. <q>A variable is a symbol that holds a place for constants.</q></cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFFeysFitch1969" class="citation book cs1"><a href="Robert_Feys" title="Robert Feys">Feys, Robert</a>; <a href="Frederic_Fitch" title="Frederic Fitch">Fitch, Frederic Brenton</a> (1969). Dictionary of symbols of mathematical logic. Amsterdam: <a href="North-Holland_Publishing_Company" class="mw-redirect" title="North-Holland Publishing Company">North-Holland Pub. Co</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/67030883">67030883</a>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFShapiroKouri_Kissel2024" class="citation cs2">Shapiro, Stewart; Kouri Kissel, Teresa (2024), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-classical/#BuilBloc">"Classical Logic"</a>, in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved September 1, 2024</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <a rel="nofollow" class="external text" href="http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html">Diophantine Equations</a>. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg <a href="https://archive.org/details/bub_gb_7DDQAAAAMAAJ/page/n472" class="extiw external" title="iarchive:bub gb 7DDQAAAAMAAJ/page/n472">456</a>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFFraleigh1989">Fraleigh 1989</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinabs0000fral/page/276/mode/2up">276</a>.

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFSorell2000">Sorell 2000</a>, p. 19.

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=moM9AQAAIAAJ">Scientific American</a>. Munn & Company. September 3, 1887. p. 148.</cite>

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<li id="cite_note-E004-19"><a href="#cite_ref-E004_19-0">^</a> Edwards Art. 4

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFHosch2010">Hosch 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ad0P0elU1_0C&pg=PA71">71</a>.

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<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFFoerster2006">Foerster 2006</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/algebratrigonome00paul_0/page/18/mode/2up">18</a>.

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<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">"Sum"</a>. mathworld.wolfram.com. Retrieved February 14, 2022.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> Edwards Art. 5

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> Edwards Art. 6

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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<ul><li><cite id="CITEREFEdwards1892" class="citation book cs1">Edwards, Joseph (1892). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.109607">An Elementary Treatise on the Differential Calculus</a> (2nd ed.). London: MacMillan and Co.</cite></li>

<li><cite id="CITEREFFoerster2006" class="citation book cs1">Foerster, Paul A. (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/algebratrigonome00paul_0">Algebra and Trigonometry: Functions and Applications</a> (classics ed.). Upper Saddle River, NJ: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-165711-3</bdi>.</cite></li>

<li><cite id="CITEREFFraleigh1989" class="citation book cs1">Fraleigh, John B. (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinabs0000fral">A First Course in Abstract Algebra</a> (4th ed.). United States: Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-52821-3</bdi>.</cite></li>

<li><cite id="CITEREFHosch2010" class="citation book cs1">Hosch, William L., ed. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ad0P0elU1_0C">The Britannica Guide to Algebra and Trigonometry</a>. Britannica Educational Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61530-219-2</bdi>.</cite></li>

<li><cite id="CITEREFMenger1954" class="citation journal cs1"><a href="Karl_Menger" title="Karl Menger">Menger, Karl</a> (1954). "On Variables in Mathematics and in Natural Science". The British Journal for the Philosophy of Science. 5 (18). University of Chicago Press: 134–142. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbjps%2FV.18.134">10.1093/bjps/V.18.134</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/685170">685170</a>.</cite></li>

<li><cite id="CITEREFPeregrin2000" class="citation book cs1"><a href="Jaroslav_Peregrin" title="Jaroslav Peregrin">Peregrin, Jaroslav</a> (2000). <a rel="nofollow" class="external text" href="http://w.jarda.peregrin.cz/mybibl/PDFTxt/414.pdf">"Variables in Natural Language: Where do they come from?"</a> (PDF). In Böttner, Michael; Thümmel, Wolf (eds.). Variable-Free Semantics. Osnabrück Secolo. pp. 46–65. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-929979-53-4</bdi>.</cite></li>

<li><cite id="CITEREFQuine1960" class="citation journal cs1"><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine, Willard V.</a> (1960). <a rel="nofollow" class="external text" href="https://voices.uchicago.edu/wittgenstein/files/2008/05/quine-variables-explained-away.pdf">"Variables Explained Away"</a> (PDF). Proceedings of the American Philosophical Society. 104 (3). American Philosophical Society: 343–347. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/985250">985250</a>.</cite></li>

<li><cite id="CITEREFSorell2000" class="citation book cs1"><a href="Tom_Sorell" title="Tom Sorell">Sorell, Tom</a> (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EksSDAAAQBAJ">Descartes: A Very Short Introduction</a>. New York: Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-285409-4</bdi>.</cite></li>

<li><cite id="CITEREFStoverWeisstein" class="citation encyclopaedia cs1">Stover, Christopher; <a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Variable.html">"Variable"</a>. In Weisstein, Eric W. (ed.). Wolfram MathWorld. Wolfram Research. Retrieved November 22, 2021.</cite></li>

<li><cite id="CITEREFTabak2014" class="citation book cs1">Tabak, John (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h-zRieb7VbwC">Algebra: Sets, Symbols, and the Language of Thought</a>. Infobase Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8160-6875-3</bdi>.</cite></li></ul>

</div>

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</style><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom" title="Axiom">Axiom</a>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>

<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)  and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>

<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>

<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>

<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>

<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>

<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>

<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>

<li><a href="Consistency" title="Consistency">Consistency</a>

<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>

<li><a href="Argument" title="Argument">Argument</a></li>

<li><a href="Soundness" title="Soundness">Soundness</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>

<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>

<li><a href="Truth_table" title="Truth table">Truth tables</a></li>

<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>

<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>

<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>

<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>

<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>

<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>

<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>

<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>

<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>

<li>Set operations:

<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>

<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Power_set" title="Power set">power set</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li>

<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>

<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>

<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>

<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>

<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>

<li><a href="Codomain" title="Codomain">codomain</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>

<li><a href="Injective_function" title="Injective function">In</a>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>

<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>

<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>

<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>

<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>

<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>

<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>

<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>

<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>

<li><a href="General_set_theory" title="General set theory">General</a></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>

<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>

<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>

<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems">list</a>), <a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>

<li><a href="Arity" title="Arity">Arity</a></li>

<li><a href="Automata_theory" title="Automata theory">Automata</a></li>

<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>

<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>

<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>

<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>

<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>

<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>

<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>

<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>

<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>

<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>

<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>

<li><a href="Formal_language" title="Formal language">Language</a></li>

<li><a href="Metalanguage" title="Metalanguage">Metalanguage</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connective</a>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>

<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>

<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>

<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>

<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>

<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>

<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>

<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>

<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>

<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>

<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>

</ul></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic systems</a> (<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:

<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>

<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>

<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>

<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>

<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>

<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>

<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>

<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>

<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>

<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>

<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>

<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:

<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:

<ul><li><a href="Euclid's_Elements" title="Euclid's Elements">Elements</a></li>

<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>

<ul><li><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>

<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Formal_system" title="Formal system">Systems</a>

<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>

<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>

<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>

<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>

<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>

<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>

<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>

<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>

<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>

<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>

<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>

<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>

<li><a href="Saturated_model" title="Saturated model">saturated</a></li>

<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>

<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>

<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>

<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>

<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>

<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>

<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>

<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>

<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>

<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>

<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>

<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>

<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>

<li><a href="T-schema" title="T-schema">T-schema</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>

<li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>

<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>

<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>

<li><a href="Computable_function" title="Computable function">Computable function</a></li>

<li><a href="Computable_set" title="Computable set">Computable set</a></li>

<li><a href="Decision_problem" title="Decision problem">Decision problem</a>

<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>

<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>

<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>

<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Concrete_category" title="Concrete category">Concrete</a>/<a href="Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li>

<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>

<li><a href="History_of_logic" title="History of logic">History of logic</a></li>

<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="Logicism" title="Logicism">Logicism</a></li>

<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Supertask" title="Supertask">Supertask</a></li></ul>

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