In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, especially as applied in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
of .
[Brown, James Robert. ]
Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures
'. United Kingdom, Taylor & Francis, 2005. 27. A
limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If is true, one can immediately deduce that is true as well, and if is false, can also be immediately deduced to be false. A
degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.
Examples
Special case examples include the following:
* All squares are
rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
s (but not all rectangles are squares); therefore the square is a special case of the rectangle. It is also a special case of the
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
.
* If an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
is defined as a triangle with ''at least'' 2 identical angles, an
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
is therefore a special case. (However, this is not true if an authority follows a different
linguistic prescription
Linguistic prescription is the establishment of rules defining publicly preferred Usage (language), usage of language, including rules of spelling, pronunciation, vocabulary, grammar, etc. Linguistic prescriptivism may aim to establish a standard ...
of an isosceles triangle having exactly 2 sides.)
*
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
, that has no solutions in positive integers with , is a special case of
Beal's conjecture, that has no primitive solutions in positive integers with , , and all greater than 2, specifically, the case of .
* The unproven
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
is a special case of the
generalized Riemann hypothesis, in the case that ''χ''(''n'') = 1 for all ''n.''
*
Fermat's little theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
a^p \equiv a \pmod p.
For example, if and , t ...
, which states "if is a prime number, then for any
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''a'', then
" is a special case of
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, then a^ is congruent to 1 modulo , where \varphi denotes Euler's totient function; that ...
, which states "if ''n'' and ''a'' are
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
positive integers, and
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
, then
", in the case that is a prime number.
*
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality
e^ + 1 = 0
where
:e is E (mathematical constant), Euler's number, the base of natural logarithms,
:i is the imaginary unit, which by definit ...
is a special case of
Euler's formula which states "for any
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
''x'':
", in the case that =
.
References
{{reflist
Mathematical logic