View source for Law (mathematics)


{{Short description|A mathematical statement which always holds true}}
{{Other uses|Law (disambiguation)}}

In [[mathematics]], a '''law''' is a [[Well-formed formula|formula]] that is always true within a given context.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Law |url=https://mathworld.wolfram.com/Law.html |access-date=2024-08-19 |website=mathworld.wolfram.com |language=en}}</ref> Laws describe a [[Relation (mathematics)|relationship]], between two or more [[expression (mathematics)|expression]]s or [[term (logic)|term]]s (which may contain [[variable (mathematics)|variable]]s), usually using [[equality (mathematics)|equality]] or [[inequality (mathematics)|inequality]],<ref>Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: https://plato.stanford.edu/entries/algebra/#Laws</ref> or between formulas themselves, for instance, in [[mathematical logic]]. For example, the formula <math>a^2 \geq 0</math> is true for all [[real number]]s ''{{Mvar|a}}'', and is therefore a law. Laws over an equality are called [[Identity (mathematics)|identities]].<ref>Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613</ref> For example, <math>(a+b)^2 = a^2 + 2ab + b^2</math>  and <math>\cos^2\theta + \sin^2\theta =1</math> are identities.<ref>{{Cite web |title=Mathwords: Identity |url=https://www.mathwords.com/i/identity.htm |access-date=2019-12-01 |website=www.mathwords.com}}</ref> Mathematical laws are distinguished from [[scientific law]]s which are based on [[observation]]s, and try to describe or [[prediction|predict]] a range of [[natural phenomena]].<ref>{{OED|law of nature}}</ref> The more significant laws are often called [[theorem]]s.

== Notable examples ==
=== Geometric laws ===
[[Triangle inequality]]: If ''{{mvar|a}}'', ''{{mvar|b}}'', and ''{{mvar|c}}'' are the lengths of the sides of a [[triangle]] then the triangle inequality states that
:<math>c \leq a + b,</math>  
with equality only in the degenerate case of a triangle with zero [[area]]. In [[Euclidean geometry]] and some other geometries, the triangle inequality is a theorem about vectors and vector lengths ([[norm (mathematics)|norm]]s):
:<math>\|\mathbf u + \mathbf v\| \leq \|\mathbf u\| + \|\mathbf v\| ,</math> 
where the length of the third side has been replaced by the length of the vector sum {{math|'''u''' + '''v'''}}. When {{math|'''u'''}} and {{math|'''v'''}} are real numbers, they can be viewed as vectors in <math>\R^1</math>, and the triangle inequality expresses a relationship between [[absolute value]]s.

[[Pythagorean theorem]]: It states that the area of the [[square]] whose side is the [[hypotenuse]] (the side opposite the [[right angle]]) is equal to the sum of the areas of the squares on the other two sides. The [[theorem]] can be written as an [[equation]] relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', sometimes called the Pythagorean equation:<ref name="Sally0">{{cite book |author1=Judith D. Sally |title=Roots to research: a vertical development of mathematical problems |author2=Paul Sally |publisher=American Mathematical Society Bookstore |year=2007 |isbn=978-0-8218-4403-8 |page=63 |chapter=Chapter 3: Pythagorean triples |chapter-url=https://books.google.com/books?id=nHxBw-WlECUC&pg=PA63}}</ref>
: <math>a^2 + b^2 = c^2 .</math>

==== Trigonometric identities ====
{{Main|List of trigonometric identities}}

Geometrically, [[trigonometric identity|trigonometric identities]] are identities involving certain functions of one or more [[angle]]s.<ref>{{Cite web |last=Stapel |first=Elizabeth |title=Trigonometric Identities |url=https://www.purplemath.com/modules/idents.htm |access-date=2019-12-01 |website=Purplemath}}</ref> They are distinct from [[Trigonometry#Triangle identities|triangle identities]], which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the [[integral|integration]] of non-trigonometric functions: a common technique which involves first using the [[Trigonometric substitution|substitution rule with a trigonometric function]], and then simplifying the resulting integral with a trigonometric identity.

One of the most prominent examples of trigonometric identities involves the equation <math>\sin^2 \theta + \cos^2 \theta = 1,</math> which is true for all [[real number|real]] values of <math>\theta</math>. On the other hand, the equation

: <math>\cos\theta = 1</math>

is only true for certain values of <math>\theta</math>, not all. For example, this equation is true when <math>\theta = 0,</math> but false when <math>\theta = 2</math>.

Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity <math>\sin(2\theta) = 2\sin\theta \cos\theta</math>, the addition formula for <math>\tan(x + y)</math>), which can be used to break down expressions of larger angles into those with smaller constituents.

=== Algebraic laws ===
[[Cauchy–Schwarz inequality]]: An upper bound on the [[Inner product space|inner product]] between two [[Vector (mathematics and physics)|vectors]] in an inner product space in terms of the product of the vector [[norm (mathematics)|norm]]s. It is considered one of the most important and widely used [[Inequality (mathematics)|inequalities]] in mathematics.<ref name="Steele">{{cite book |last=Steele |first=J. Michael |url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html |title=The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities |publisher=The Mathematical Association of America |year=2004 |isbn=978-0521546775 |page=1 |quote=...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.}}</ref>

The Cauchy–Schwarz inequality states that for all vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> of an [[inner product space]]
:<math>\left \vert \langle \bold u, \bold v \rangle \right \vert \leq \langle \bold u, \bold u \rangle \cdot \langle \bold v, \bold v \rangle</math>
where <math>\langle \cdot, \cdot \rangle</math>
is the [[inner product]]. Examples of inner products include the real and complex [[dot product]]; see the [[Inner product space#Some examples|examples in inner product]]. Every inner product gives rise to a Euclidean <math>l_2</math> [[Norm (mathematics)|norm]], called the {{em|canonical}} or [[Inner product space#Norm|{{em|induced}} {{em|norm}}]], where the norm of a vector <math>\mathbf{u}</math> is denoted and defined by
:<math>\|\mathbf{u}\| := \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle},</math> 
where <math>\langle \mathbf{u}, \mathbf{u} \rangle</math> is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:<ref name="Strang5">{{cite book |last=Strang |first=Gilbert |title=Linear Algebra and its Applications |date=19 July 2005 |publisher=Cengage Learning |isbn=978-0030105678 |edition=4th |location=Stamford, CT |pages=154–155 |chapter=3.2}}</ref><ref name=":0">{{cite book |last1=Hunter |first1=John K. |url=https://books.google.com/books?id=oOYQVeHmNk4C |title=Applied Analysis |last2=Nachtergaele |first2=Bruno |publisher=World Scientific |year=2001 |isbn=981-02-4191-7}}</ref>
:<math>\left \vert \langle \bold u, \bold v \rangle \right \vert \leq \langle \bold u, \bold u \rangle \cdot \langle \bold v, \bold v \rangle</math>
Moreover, the two sides are equal if and only if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are [[Linear independence|linearly dependent]].<ref>{{cite book |last1=Bachmann |first1=George |url=https://books.google.com/books?id=PkHhBwAAQBAJ |title=Fourier and Wavelet Analysis |last2=Narici |first2=Lawrence |last3=Beckenstein |first3=Edward |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=9781461205050 |page=14}}</ref><ref>{{cite book |last=Hassani |first=Sadri |title=Mathematical Physics: A Modern Introduction to Its Foundations |publisher=Springer |year=1999 |isbn=0-387-98579-4 |page=29 |quote=Equality holds iff <c{{pipe}}c>=0 or {{pipe}}c>=0. From the definition of {{pipe}}c>, we conclude that {{pipe}}a> and {{pipe}}b> must be proportional.}}</ref><ref>{{cite book |last1=Axler |first1=Sheldon |url=https://books.google.com/books?id=CQWwoQEACAAJ |title=Linear Algebra Done Right, 3rd Ed. |date=2015 |publisher=Springer International Publishing |isbn=978-3-319-11079-0 |page=172 |quote=This inequality is an equality if and only if one of u, v is a scalar multiple of the other.}}</ref>

=== Combinatorial laws ===
[[Pigeonhole principle]]: If ''{{mvar|n}}'' items are put into ''{{mvar|m}}'' containers, with {{math|''n'' > ''m''}}, then at least one container must contain more than one item.<ref name="Herstein64">{{harvnb|Herstein|1964|loc=p.&nbsp;90}}</ref> For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into.

=== Logical laws ===
[[De Morgan's laws]]:  In [[Propositional calculus|propositional logic]] and [[Boolean algebra]], '''De Morgan's laws''',<ref>{{Cite book |last1=Copi |first1=Irving M. |url=https://www.taylorfrancis.com/books/mono/10.4324/9781315510897/introduction-logic-irving-copi-carl-cohen-kenneth-mcmahon |title=Introduction to Logic |last2=Cohen |first2=Carl |last3=McMahon |first3=Kenneth |date=2016 |isbn=9781315510880 |doi=10.4324/9781315510897}}</ref><ref>{{citation |last=Hurley |first=Patrick J. |title=A Concise Introduction to Logic |year=2015 |edition=12th |publisher=Cengage Learning |isbn=978-1-285-19654-1}}</ref><ref>{{Cite book |last=Moore |first=Brooke Noel |url=https://www.worldcat.org/oclc/689858599 |title=Critical thinking |date=2012 |publisher=McGraw-Hill |others=Richard Parker |isbn=978-0-07-803828-0 |edition=10th |location=New York |oclc=689858599}}</ref> also known as '''De Morgan's theorem''',<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/DeMorgan.html DeMorgan's Theorem]{{sic}}</ref> are a pair of transformation rules that are both [[Validity (logic)|valid]] [[Rule of inference|rules of inference]]. They are named after [[Augustus De Morgan]], a 19th-century British mathematician. The rules allow the expression of [[Logical conjunction|conjunctions]] and [[Logical disjunction|disjunctions]] purely in terms of each other via [[Logical negation|negation]]. The rules can be expressed in English as:
* not (A or B) = (not A) and (not B)
* not (A and B) = (not A) or (not B)  where "A or B" is an "[[inclusive or]]" meaning ''at least'' one of A or B rather than an "[[exclusive or]]" that means ''exactly'' one of A or B. In [[formal language]], the rules are written as <math display="block"> \begin{align}
 \neg(P\lor Q) &\iff (\neg P) \land (\neg Q), \quad \text{ and } \\
 \neg(P\land Q) &\iff (\neg P) \lor (\neg Q),
\end{align}</math> where ''P'' and ''Q ''are [[Proposition|propositions]],
* [[Negation|<math>\neg</math>]] is the negation logic operator (NOT),
* [[Logical conjunction|<math>\land</math>]] is the conjunction logic operator (AND),
* [[Logical disjunction|<math>\lor</math>]] is the disjunction logic operator (OR),
* [[If and only if|<math>\iff</math>]] is a [[Metalogic|metalogical]] symbol meaning "can be replaced in a [[Formal proof|logical proof]] with", often read as "if and only if". For any combination of true/false values for P and Q, the left and right sides of the arrow will hold the same truth value after evaluation.

The three [[Law of thought|Laws of thought]] are:
* The [[law of identity]]: 'Whatever is, is.'<ref name="Russell 1912:72,1997 edition">Russell 1912:72,1997 edition.</ref> For all a: a = a.
* The [[law of non-contradiction]] (alternately the 'law of contradiction'<ref name="Russell 1912:72, 1997 edition">Russell 1912:72, 1997 edition.</ref>): 'Nothing can both be and not be.'<ref name="Russell 1912:72,1997 edition" />
* The law of excluded middle: 'Everything must either be or not be.'<ref name="Russell 1912:72,1997 edition" />  In accordance with the [[law of excluded middle]] or excluded third, for every proposition, either its positive or negative form is true: A[[Logical disjunction|∨]]¬A.

=== Phinominological laws ===
{{See also|Mathematical coincidence|Empirical statistical laws}}
[[Benford's law]] is an observation that in many real-life sets of numerical [[data]], the [[leading digit]] is likely to be small.<ref name="BergerHill2011">Arno Berger and Theodore P. Hill, [http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1074&context=rgp_rsr Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem], 2011.</ref> In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. Uniformly distributed digits would each occur about 11.1% of the time.<ref>{{cite web |author=Weisstein, Eric W. |title=Benford's Law |url=http://mathworld.wolfram.com/BenfordsLaw.html |access-date=7 June 2015 |website=MathWorld, A Wolfram web resource}}</ref>

[[Strong law of small numbers]], in a humorous way, states any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few.

== See also ==

* [[Formula]]
* [[List of inequalities]]
* [[List of mathematical identities]]
* [[List of laws]]
* [[Statement (logic)]]
* [[Tautology (logic)]]

== Citations ==
{{reflist}}

==References==
*[[Bertrand Russell]], ''The Problems of Philosophy'' (1912), Oxford University Press, New York, 1997, {{ISBN|0-19-511552-X}}.
* {{ citation | first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = Blaisdell Publishing Company | location = Waltham }}

== External links ==

* [https://web.archive.org/web/20190612171441/https://encyclopedia-of-equation.webnode.jp/ The Encyclopedia of Equation]   Online encyclopedia of mathematical identities (archived)
* [http://sites.google.com/site/tpiezas/Home/ A Collection of Algebraic Identities]


[[Category:Mathematical terminology]]
[[Category:Theorems]]
[[Category:Mathematical logic]]