In mathematics, a law is a formula that is always true within a given context. Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, or between formulas themselves, for instance, in mathematical logic. For example, the formula
a
2
≥
0
{\displaystyle a^{2}\geq 0}
is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. For example,
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
and
cos
2
θ
+
sin
2
θ
=
1
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
are identities. Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena.
In modern mathematics, the term is rarely used, except for historical terminology, such as in the law of sines. The terms of "theorem" or "property" are generally preferred, as more accurate.
Contents
Notable examples
Geometric laws
Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that
c
≤
a
+
b
,
{\displaystyle 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):
‖
u
+
v
‖
≤
‖
u
‖
+
‖
v
‖
,
{\displaystyle \|\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where the length of the third side has been replaced by the length of the vector sum u + v. When u and v are real numbers, they can be viewed as vectors in
R
1
{\displaystyle \mathbb {R} ^{1}}
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.
The Cauchy–Schwarz inequality states that for all vectors
u
{\displaystyle \mathbf {u} }
and
v
{\displaystyle \mathbf {v} }
of an inner product space
|
⟨
u
,
v
⟩
|
≤
⟨
u
,
u
⟩
⋅
⟨
v
,
v
⟩
{\displaystyle \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
Combinatorial laws
Pigeonhole principle: If n items are put into m containers, with n > m, then at least one container must contain more than one item. 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 logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, 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
¬
(
P
∨
Q
)
⟺
(
¬
P
)
∧
(
¬
Q
)
,
and
¬
(
P
∧
Q
)
⟺
Phenomenological laws
Benford's law is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. 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.
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.