special case

{{Short description|Specific, usually well-known application of a mathematical rule or law}}

In logic, especially as applied in mathematics, concept {{mvar|A}} is a special case or specialization of concept {{mvar|B}} precisely if every instance of {{mvar|A}} is also an instance of {{mvar|B}} but not vice versa, or equivalently, if {{mvar|B}} is a generalization of {{mvar|A}}.Brown, James Robert. [https://books.google.com/books?id=fmXR2P0Ta7AC&pg=PA27 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 {{Mvar|B}} is true, one can immediately deduce that {{Mvar|A}} is true as well, and if {{Mvar|B}} is false, {{Mvar|A}} 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 rectangles (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.
If an isosceles triangle is defined as a triangle with at least 2 identical angles, an equilateral triangle is therefore a special case. (However, this is not true if an authority follows a different linguistic prescription of an isosceles triangle having exactly 2 sides.)
Fermat's Last Theorem, that {{mvar|aⁿ + bⁿ {{=}} cⁿ}} has no solutions in positive integers with {{mvar|n > 2}}, is a special case of Beal's conjecture, that {{mvar|a^x + b^y {{=}} c^z}} has no primitive solutions in positive integers with {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} all greater than 2, specifically, the case of {{mvar|x {{=}} y {{=}} z}}.
The unproven Riemann hypothesis is a special case of the generalized Riemann hypothesis, in the case that χ(n) = 1 for all n.
Fermat's little theorem, which states "if {{Mvar|p}} is a prime number, then for any integer a, then $a^p \equiv a \pmod p$ " is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and $\phi(n)$ is Euler's totient function, then $a^{\varphi (n)} \equiv 1 \pmod{n}$ ", in the case that {{Mvar|n}} is a prime number.
Euler's identity $e^{i \pi} = -1$ is a special case of Euler's formula which states "for any real number x: $e^{ix} = \cos x + i\sin x$ ", in the case that {{Mvar|x}} = $\pi$ .

References

Category:Mathematical logic