{{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}}.<ref>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.</ref> 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|a<sup>n</sup> + b<sup>n</sup> {{=}} c<sup>n</sup>}} has no solutions in positive integers with {{mvar|n > 2}}, is a special case of Beal's conjecture, that {{mvar|a<sup>x</sup> + b<sup>y</sup> {{=}} c<sup>z</sup>}} 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 <math>a^p \equiv a \pmod p</math>" is a special case of Euler's theorem, which states "if ''n'' and ''a'' are coprime positive integers, and <math>\phi(n)</math> is Euler's totient function, then <math>a^{\varphi (n)} \equiv 1 \pmod{n}</math>", in the case that {{Mvar|n}} is a prime number. * Euler's identity <math>e^{i \pi} = -1</math> is a special case of Euler's formula which states "for any real number ''x'': <math>e^{ix} = \cos x + i\sin x</math>", in the case that {{Mvar|x}} = <math>\pi</math>.
==References== {{reflist}} Category:Mathematical logic