Proportionality vs. Linearity - Revision history

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[[File:Proportional variables.svg|thumb|300x300px|The variable ''y'' is directly proportional to the variable ''x'' with proportionality constant ~0.6.]]
[[File:Inverse proportionality function plot.gif|thumb|300x300px|The variable ''y'' is inversely proportional to the variable ''x'' with proportionality constant 1.]]

In mathematics, two varying quantities are said to be in a relation of '''proportionality''', multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the '''coefficient of proportionality''' or '''proportionality constant'''.
*If the ''ratio'' (<math>\frac{y}{x}</math>) of two variables ({{mvar|x}} and {{mvar|y}}) is equal to a constant (''k'' = <math>\frac{y}{x}</math>), then the variable in the numerator of the ratio ({{mvar|y}}) can be product of the other variable and the constant {{math|1=(''y'' = ''k'' ⋅ ''x'')}}. In this case {{mvar|y}} is said to be ''directly proportional'' to {{mvar|x}} with proportionality constant {{mvar|k}}. Equivalently one may write <math> x = \frac{1}{k}</math>⋅ ''y''; that is, {{mvar|x}} is directly proportional to {{mvar|y}} with proportionality constant <math>\frac{1}{k} = (\frac{x}{y})</math>. If the term ''proportional'' is connected to two variables without further qualification, generally direct proportionality can be assumed.
*If the ''product'' of two variables {{math|(''x'' ⋅ ''y'')}} is equal to a constant {{math|1=(''k'' = ''x'' ⋅ ''y'')}}, then the two are said to be ''inversely proportional'' to each other with the proportionality constant {{mvar|k}}. Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant {{mvar|k}} (<math>x</math> = ''k'' ⋅ <math>\frac{1}{y}</math> and {{math|1=''y'' = ''k'' ⋅ }}<math>\frac{1}{x} </math>).

If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a '''proportion''', e.g., <math>\frac{a}{b} = \frac{x}{y} </math> = ⋯ = ''k'' (for details see Ratio).

==Direct proportionality ==
Given two variables ''x'' and ''y'', ''y'' is '''directly proportional''' to ''x'' if there is a non-zero constant ''k'' such that

: <math>y = kx.</math>

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~":
: <math>y \propto x,</math>&emsp;or&emsp;<math>y \sim x.</math>

For <math>x \ne 0</math> the '''proportionality constant''' can be expressed as the ratio

: <math> k = \frac{y}{x}.</math>

It is also called the '''constant of variation''' or '''constant of proportionality'''.

A direct proportionality can also be viewed as a linear equation in two variables with a ''y''-intercept of 0 and a slope of ''k''. This corresponds to linear growth.

===Examples===
* If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
* The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
* On a map of a sufficiently small geographical area, drawn to scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.
* The force, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational acceleration.
* The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.

== Inverse proportionality ==
[[File:Inverse proportionality function plot.gif|thumb|300x300px|Inverse proportionality with a function of ''y'' = 1/''x'']]

The concept of ''inverse proportionality'' can be contrasted with ''direct proportionality''. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality ''k'') is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel.

Formally, two variables are '''inversely proportional''' (also called '''varying inversely''', in '''inverse variation''', in '''inverse proportion''', in '''reciprocal proportion''') if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable ''y'' is inversely proportional to the variable ''x'' if there exists a non-zero constant ''k'' such that
: <math>y = \frac{k}{x},</math>
or equivalently, <math>xy = k.</math> Hence the constant "''k''" is the product of ''x'' and ''y''.

The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the ''x'' and ''y'' values of each point on the curve equals the constant of proportionality (''k''). Since neither ''x'' nor ''y'' can equal zero (because ''k'' is non-zero), the graph never crosses either axis.

==Hyperbolic coordinates==
The concepts of ''direct'' and ''inverse'' proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

==Linearity==
'''Linearity''' is the property of a mathematical relationship (''[[function (mathematics)|function]]'') that can be [[graph of a function|graphically]] represented as a straight [[Line (geometry)|line]]. Linearity is closely related to [[Proportionality (mathematics)|proportionality]]. Examples in [[physics]] include the linear relationship of [[voltage]] and [[Electric current|current]] in an [[electrical conductor]] ([[Ohm's law]]), and the relationship of [[mass]] and [[weight]]. By contrast, more complicated relationships are [[nonlinear]].

Generalized for functions in more than one [[dimension (mathematics)|dimension]], linearity means the property of a function of being compatible with [[addition]] and [[scale analysis (mathematics)|scaling]], also known as the [[superposition principle]].

The word '''linear''' comes from [[Latin]] ''linearis'', "pertaining to or resembling a line".

==In mathematics==
In mathematics, a [[linear map]] or [[linear function]] ''f''(''x'') is a function that satisfies the two properties:<ref>{{cite book|author=Edwards, Harold M.|title=Linear Algebra|publisher=Springer|year=1995|isbn=9780817637316|page=78|url=https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78}}</ref>

* [[Additive map|Additivity]]: {{nowrap|1=''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')}}.
* [[Homogeneous function|Homogeneity]] of degree 1: {{nowrap|1=''f''(α''x'') = α ''f''(''x'')}} for all α.

These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a [[real number]], but can in general be an [[element (mathematics)|element]] of any [[vector space]]. A more special definition of [[linear function#As a polynomial function|linear function]], not coinciding with the definition of linear map, is used in elementary mathematics (see below).

Additivity alone implies homogeneity for [[Rational number|rational]] α, since <math>f(x+x)=f(x)+f(x)</math> implies <math>f(nx)=n f(x)</math> for any [[natural number]] ''n'' by [[mathematical induction]], and then <math>n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x)</math> implies <math>f(\tfrac{n}{m}x) = \tfrac{n}{m} f(x)</math>. The [[Dense set|density]] of the rational numbers in the reals implies that any additive [[continuous function]] is homogeneous for any real number α, and is therefore linear.

The concept of linearity can be extended to linear [[Operator (mathematics)|operators]]. Important examples of linear operators include the [[derivative]] considered as a [[differential operator]], and other operators constructed from it, such as [[del]] and the [[Laplacian]]. When a [[differential equation]] can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

[[Linear algebra]] is the branch of mathematics concerned with the study of [[vector (mathematics)|vectors]], [[vector space]]s (also called 'linear spaces'), [[linear transformation]]s (also called 'linear maps'), and systems of linear equations.

For a description of linear and nonlinear equations, see ''[[linear equation]]''.

===Linear polynomials===
In a different usage to the above definition, a [[polynomial]] of degree 1 is said to be linear, because the [[graph of a function]] of that form is a straight line.<ref>[[James Stewart (mathematician)|Stewart, James]] (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. {{isbn|978-0-495-01166-8}}, Section 1.2</ref>

Over the reals, a [[linear equation]] is one of the forms:

:<math>f(x) = m x + b\ </math>

where ''m'' is often called the [[slope]] or [[gradient]]; ''b'' the [[y-intercept]], which gives the point of intersection between the graph of the function and the ''y''-axis.

Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so [[if and only if]] {{nowrap|1=''b'' = 0}}. Hence, if {{nowrap|''b'' ≠ 0}}, the function is often called an '''affine function''' (see in greater generality [[affine transformation]]).

===Boolean functions===
[[File:Hasse-linear.svg|thumb|right|Hasse diagram of a linear Boolean function]]
In [[Boolean algebra (logic)|Boolean algebra]], a linear function is a function <math>f</math> for which there exist <math>a_0, a_1, \ldots, a_n \in \{0,1\}</math> such that
:<math>f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)</math>, where <math>b_1, \ldots, b_n \in \{0,1\}.</math>

Note that if <math>a_0 = 1</math>, the above function is considered affine in linear algebra (i.e. not linear).

A Boolean function is linear if one of the following holds for the function's [[truth table]]:
# In every row in which the truth value of the function is [[Truth value#Classical logic|T]], there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, {{nowrap|1=''f''(F, F, ..., F) = F}}, and these functions correspond to linear maps over the Boolean vector space.
# In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the [[truth value]] of the function is F, there are an odd number of Ts assigned to arguments. In this case, ''f''(F, F, ..., F) = T.

Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.

Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.

==References==
#Ya. B. Zeldovich, I. M. Yaglom: Higher math for beginners, p. 34–35.
#Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, p. 85–101.
#Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.
#Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.
#Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.

← Older revision		Revision as of 04:25, 31 October 2021
Line 94:		Line 94:

	Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.		Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.
−
−	~~==References==~~
−	~~#Ya. B. Zeldovich, I. M. Yaglom: Higher math for beginners, p. 34–35.~~
−	~~#Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, p. 85–101.~~
−	~~#Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.~~
−	~~#Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.~~
−	#Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.
−	~~# Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316.~~
−	~~# Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2~~

	== Licensing ==		== Licensing ==

← Older revision		Revision as of 19:48, 24 October 2021
Line 103:		Line 103:
	# Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316.		# Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316.
	# Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2		# Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Proportionality_(mathematics) Proportionality (mathematics), Wikipedia] under a CC BY-SA license
		+	* [https://en.wikipedia.org/wiki/Linearity Linearity, Wikipedia] under a CC BY-SA license

@@ Line 1: / Line 1: @@
+==Propotionality==
 [[File:Proportional variables.svg|thumb|300x300px|The variable ''y'' is directly proportional to the variable ''x'' with proportionality constant ~0.6.]]
 [[File:Inverse proportionality function plot.gif|thumb|300x300px|The variable ''y'' is inversely proportional to the variable ''x'' with proportionality constant 1.]]
@@ Line 9: / Line 9: @@
 If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a '''proportion''', e.g., <math>\frac{a}{b} = \frac{x}{y} </math> = ⋯ = ''k'' (for details see Ratio).
-==Direct proportionality ==
+===Direct proportionality ===
 Given two variables ''x'' and ''y'', ''y'' is '''directly proportional''' to ''x'' if there is a non-zero constant ''k'' such that
@@ Line 25: / Line 25: @@
 A direct proportionality can also be viewed as a linear equation in two variables with a ''y''-intercept of 0 and a slope of ''k''. This corresponds to linear growth.
-===Examples===
+====Examples====
 * If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
 * The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
@@ Line 32: / Line 32: @@
 * The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.
-== Inverse proportionality ==
+=== Inverse proportionality ===
 [[File:Inverse proportionality function plot.gif|thumb|300x300px|Inverse proportionality with a function of ''y'' = 1/''x'']]
@@ Line 43: / Line 43: @@
 The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the ''x'' and ''y'' values of each point on the curve equals the constant of proportionality (''k''). Since neither ''x'' nor ''y'' can equal zero (because ''k'' is non-zero), the graph never crosses either axis.
-==Hyperbolic coordinates==
+===Hyperbolic coordinates===
 The concepts of ''direct'' and ''inverse'' proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.
@@ Line 53: / Line 53: @@
 The word '''linear''' comes from Latin ''linearis'', "pertaining to or resembling a line".
-==In mathematics==
+===In mathematics===
 In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties:
@@ Line 69: / Line 69: @@
 For a description of linear and nonlinear equations, see ''linear equation''.
-===Linear polynomials===
+====Linear polynomials====
 In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.
@@ Line 80: / Line 80: @@
 Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if ''b'' = 0. Hence, if ''b'' ≠ 0, the function is often called an '''affine function''' (see in greater generality affine transformation).
-===Boolean functions===
+====Boolean functions====
 [[File:Hasse-linear.svg|thumb|right|Hasse diagram of a linear Boolean function]]
 In Boolean algebra, a linear function is a function <math>f</math> for which there exist <math>a_0, a_1, \ldots, a_n \in \{0,1\}</math> such that

@@ Line 47: / Line 47: @@
 ==Linearity==
-'''Linearity''' is the property of a mathematical relationship (''[[function (mathematics)|function]]'') that can be [[graph of a function|graphically]] represented as a straight [[Line (geometry)|line]]. Linearity is closely related to [[Proportionality (mathematics)|proportionality]]. Examples in [[physics]] include the linear relationship of [[voltage]] and [[Electric current|current]] in an [[electrical conductor]] ([[Ohm's law]]), and the relationship of [[mass]] and [[weight]]. By contrast, more complicated relationships are [[nonlinear]].
+'''Linearity''' is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.
-Generalized for functions in more than one [[dimension (mathematics)|dimension]], linearity means the property of a function of being compatible with [[addition]] and [[scale analysis (mathematics)|scaling]], also known as the [[superposition principle]].
+Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
-The word '''linear''' comes from [[Latin]] ''linearis'', "pertaining to or resembling a line".
+The word '''linear''' comes from Latin ''linearis'', "pertaining to or resembling a line".
 ==In mathematics==
-In mathematics, a [[linear map]] or [[linear function]] ''f''(''x'') is a function that satisfies the two properties:<ref>{{cite book|author=Edwards, Harold M.|title=Linear Algebra|publisher=Springer|year=1995|isbn=9780817637316|page=78|url=https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78}}</ref>
+In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties:
-* [[Additive map|Additivity]]: {{nowrap|1=''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')}}.
+* Additivity: ''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'').
-* [[Homogeneous function|Homogeneity]] of degree 1: {{nowrap|1=''f''(α''x'') = α ''f''(''x'')}} for all α.
+* Homogeneity of degree 1: ''f''(α''x'') = α ''f''(''x'') for all α.
-These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a [[real number]], but can in general be an [[element (mathematics)|element]] of any [[vector space]]. A more special definition of [[linear function#As a polynomial function|linear function]], not coinciding with the definition of linear map, is used in elementary mathematics (see below).
+These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
-Additivity alone implies homogeneity for [[Rational number|rational]] α, since <math>f(x+x)=f(x)+f(x)</math> implies <math>f(nx)=n f(x)</math> for any [[natural number]] ''n'' by [[mathematical induction]], and then <math>n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x)</math> implies <math>f(\tfrac{n}{m}x) = \tfrac{n}{m} f(x)</math>. The [[Dense set|density]] of the rational numbers in the reals implies that any additive [[continuous function]] is homogeneous for any real number α, and is therefore linear.
+Additivity alone implies homogeneity for rational α, since <math>f(x+x)=f(x)+f(x)</math> implies <math>f(nx)=n f(x)</math> for any natural number ''n'' by mathematical induction, and then <math>n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x)</math> implies <math>f(\tfrac{n}{m}x) = \tfrac{n}{m} f(x)</math>. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.
-The concept of linearity can be extended to linear [[Operator (mathematics)|operators]]. Important examples of linear operators include the [[derivative]] considered as a [[differential operator]], and other operators constructed from it, such as [[del]] and the [[Laplacian]]. When a [[differential equation]] can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
+The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
-[[Linear algebra]] is the branch of mathematics concerned with the study of [[vector (mathematics)|vectors]], [[vector space]]s (also called 'linear spaces'), [[linear transformation]]s (also called 'linear maps'), and systems of linear equations.
+Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called 'linear spaces'), linear transformations (also called 'linear maps'), and systems of linear equations.
-For a description of linear and nonlinear equations, see ''[[linear equation]]''.
+For a description of linear and nonlinear equations, see ''linear equation''.
 ===Linear polynomials===
-In a different usage to the above definition, a [[polynomial]] of degree 1 is said to be linear, because the [[graph of a function]] of that form is a straight line.<ref>[[James Stewart (mathematician)|Stewart, James]] (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. {{isbn|978-0-495-01166-8}}, Section 1.2</ref>
+In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.
-Over the reals, a [[linear equation]] is one of the forms:
+Over the reals, a linear equation is one of the forms:
 :<math>f(x) = m x + b\ </math>
-where ''m'' is often called the [[slope]] or [[gradient]]; ''b'' the [[y-intercept]], which gives the point of intersection between the graph of the function and the ''y''-axis.
+where ''m'' is often called the slope or gradient; ''b'' the y-intercept, which gives the point of intersection between the graph of the function and the ''y''-axis.
-Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so [[if and only if]] {{nowrap|1=''b'' = 0}}. Hence, if {{nowrap|''b'' ≠ 0}}, the function is often called an '''affine function''' (see in greater generality [[affine transformation]]).
+Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if ''b'' = 0. Hence, if ''b'' ≠ 0, the function is often called an '''affine function''' (see in greater generality affine transformation).
 ===Boolean functions===
 [[File:Hasse-linear.svg|thumb|right|Hasse diagram of a linear Boolean function]]
-In [[Boolean algebra (logic)|Boolean algebra]], a linear function is a function <math>f</math> for which there exist <math>a_0, a_1, \ldots, a_n \in \{0,1\}</math> such that
+In Boolean algebra, a linear function is a function <math>f</math> for which there exist <math>a_0, a_1, \ldots, a_n \in \{0,1\}</math> such that
 :<math>f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)</math>, where <math>b_1, \ldots, b_n \in \{0,1\}.</math>
 Note that if <math>a_0 = 1</math>, the above function is considered affine in linear algebra (i.e. not linear).
-A Boolean function is linear if one of the following holds for the function's [[truth table]]:
+A Boolean function is linear if one of the following holds for the function's truth table:
-# In every row in which the truth value of the function is [[Truth value#Classical logic|T]], there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, {{nowrap|1=''f''(F, F, ..., F) = F}}, and these functions correspond to linear maps over the Boolean vector space.
+# In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, ''f''(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space.
-# In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the [[truth value]] of the function is F, there are an odd number of Ts assigned to arguments. In this case, ''f''(F, F, ..., F) = T.
+# In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, ''f''(F, F, ..., F) = T.
 Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.
@@ Line 101: / Line 101: @@
 #Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.
 #Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.