Law (mathematics)

In mathematics, a law is a formula that is always true within a certain context.^[1] Laws describe a relationship, between two or more terms or exprssions (which may contain variables), usually using equality or inequality,^[2] or between formulas themselves, for instace, in mathematical logic. For example, the formula $a^{2}\geq 0$ is true for all real-numbers $a$ , and is therefor a law. Laws over an equality are called indentities.^[3] For example, $(a+b)^{2}=a^{2}+2ab+b^{2}$ and $\cos ^{2}\theta +\sin ^{2}\theta =1$ are identities.^[4] Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena.^[5] The more significant laws are often called theorems.

Notable Examples

Geometric laws

Triange inequality: If $a$ , $b$ , and $c$ are the lengths of the sides of a triangle then the triangle inequality states that

c\leq a+b,

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 (norms):

\|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,

where the length of the third side has been replaced by the length of the vector sum $u + v$ . When $u$ and $v$ are real numbers, they can be viewed as vectors in $\mathbb {R} ^{1}$ , and the triangle inequality expresses a relationship between absolute values.

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:^[6]
$a^{2}+b^{2}=c^{2}.$

Trigonometric identities

Geometrically, trigonometric identities are identities involving certain functions of one or more angles.^[7] They are distinct from 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 integration of non-trigonometric functions: a common technique which involves first using the 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 $\sin ^{2}\theta +\cos ^{2}\theta =1,$ which is true for all real values of $\theta$ . On the other hand, the equation

\cos \theta =1

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

Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity $\sin(2\theta )=2\sin \theta \cos \theta$ , the addition formula for $\tan(x+y)$ ), 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 between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.^[8]

The Cauchy–Schwarz inequality states that for all vectors $\mathbf {u}$ and $\mathbf {v}$ of an inner product space

\left\vert \langle {\mathbf {u}},{\mathbf {v}}\rangle \right\vert \leq \langle {\mathbf {u}},{\mathbf {u}}\rangle \cdot \langle {\mathbf {v}},{\mathbf {v}}\rangle

where $\langle \cdot ,\cdot \rangle$ is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean $l_{2}$ norm, called the canonical or induced norm, where the norm of a vector $\mathbf {u}$ is denoted and defined by

\|\mathbf {u} \|:={\sqrt {\langle \mathbf {u} ,\mathbf {u} \rangle }},

where $\langle \mathbf {u} ,\mathbf {u} \rangle$ 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:^[9]^[10]

\left\vert \langle {\mathbf {u}},{\mathbf {v}}\rangle \right\vert \leq \langle {\mathbf {u}},{\mathbf {u}}\rangle \cdot \langle {\mathbf {v}},{\mathbf {v}}\rangle

Moreover, the two sides are equal if and only if $\mathbf {u}$ and $\mathbf {v}$ are linearly dependent.^[11]^[12]^[13]

Logical Laws

De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws,^[14]^[15]^[16] also known as De Morgan's theorem,^[17] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via 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
$\neg (P\lor Q)\iff (\neg P)\land (\neg Q),$ and

$\neg (P\land Q)\iff (\neg P)\lor (\neg Q)$ where
- P and Q are propositions,
- $\neg$ is the negation logic operator (NOT),
- $\land$ is the conjunction logic operator (AND),
- $\lor$ is the disjunction logic operator (OR),
- $\iff$ is a metalogical symbol meaning "can be replaced in a 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.
Modus ponens: a deductive argument form and rule of inference.^[18] It can be summarized as "P implies Q. P is true. Therefore, Q must also be true." Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens.

The form of a modus ponens argument is a mixed hypothetical syllogism, with two premises and a conclusion:

If P, then Q.
P.
Therefore, Q.

The first premise is a conditional ("if–then") claim, namely that P implies Q. The second premise is an assertion that P, the antecedent of the conditional claim, is the case. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be the case as well.

An example of an argument that fits the form modus ponens:

If today is Tuesday, then John will go to work.
Today is Tuesday.
Therefore, John will go to work.

The three Laws of thought
- The law of identity: 'Whatever is, is.'^[19] For all a: a = a.
- The law of non-contradiction (alternately the 'law of contradiction'^[20]): 'Nothing can both be and not be.'^[19]
- The law of excluded middle: 'Everything must either be or not be.'^[19] In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A.

References

^ Weisstein, Eric W. "Law". mathworld.wolfram.com. Retrieved 2024-08-19.
^ 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
^ Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
^ "Mathwords: Identity". www.mathwords.com. Retrieved 2019-12-01.
^ "law of nature". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.
^ Stapel, Elizabeth. "Trigonometric Identities". Purplemath. Retrieved 2019-12-01.
^ Steele, J. Michael (2004). The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities. The Mathematical Association of America. p. 1. ISBN 978-0521546775. ...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
^ Strang, Gilbert (19 July 2005). "3.2". Linear Algebra and its Applications (4th ed.). Stamford, CT: Cengage Learning. pp. 154–155. ISBN 978-0030105678.
^ Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis. World Scientific. ISBN 981-02-4191-7.
^ Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. p. 14. ISBN 9781461205050.
^ Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN 0-387-98579-4. Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
^ Axler, Sheldon (2015). Linear Algebra Done Right, 3rd Ed. Springer International Publishing. p. 172. ISBN 978-3-319-11079-0. This inequality is an equality if and only if one of u, v is a scalar multiple of the other.
^ Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016). Introduction to Logic. doi:10.4324/9781315510897. ISBN 9781315510880.
^ Hurley, Patrick J. (2015), A Concise Introduction to Logic (12th ed.), Cengage Learning, ISBN 978-1-285-19654-1
^ Moore, Brooke Noel (2012). Critical thinking. Richard Parker (10th ed.). New York: McGraw-Hill. ISBN 978-0-07-803828-0. OCLC 689858599.
^ http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/DeMorgan.html DeMorgan's [sic] Theorem
^ Enderton 2001:110
^ ^a ^b ^c Russell 1912:72,1997 edition.
^ Russell 1912:72, 1997 edition.

External links

The Encyclopedia of Equation Online encyclopedia of mathematical identities (archived)
A Collection of Algebraic Identities

[1] Weisstein, Eric W. "Law". mathworld.wolfram.com. Retrieved 2024-08-19.

[2] 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

[3] Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613

[4] "Mathwords: Identity". www.mathwords.com. Retrieved 2019-12-01.

[5] "law of nature". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[Sally0-6] Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.

[7] Stapel, Elizabeth. "Trigonometric Identities". Purplemath. Retrieved 2019-12-01.

[Steele-8] Steele, J. Michael (2004). The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities. The Mathematical Association of America. p. 1. ISBN 978-0521546775. ...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.

[Strang5-9] Strang, Gilbert (19 July 2005). "3.2". Linear Algebra and its Applications (4th ed.). Stamford, CT: Cengage Learning. pp. 154–155. ISBN 978-0030105678.

[:0-10] Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis. World Scientific. ISBN 981-02-4191-7.

[11] Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. p. 14. ISBN 9781461205050.

[12] Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN 0-387-98579-4. Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.

[13] Axler, Sheldon (2015). Linear Algebra Done Right, 3rd Ed. Springer International Publishing. p. 172. ISBN 978-3-319-11079-0. This inequality is an equality if and only if one of u, v is a scalar multiple of the other.

[14] Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016). Introduction to Logic. doi:10.4324/9781315510897. ISBN 9781315510880.

[15] Hurley, Patrick J. (2015), A Concise Introduction to Logic (12th ed.), Cengage Learning, ISBN 978-1-285-19654-1

[16] Moore, Brooke Noel (2012). Critical thinking. Richard Parker (10th ed.). New York: McGraw-Hill. ISBN 978-0-07-803828-0. OCLC 689858599.

[17] ttp://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/DeMorgan.html DeMorgan's [sic] Theorem

[18] Enderton 2001:110

[Russell_1912:72,1997_edition-19] Russell 1912:72,1997 edition.

[Russell_1912:72,_1997_edition-20] Russell 1912:72, 1997 edition.

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