Special case

 ( Mathematical Logic ) 

In logic, especially as applied in mathematics, 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 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.

• All squares are (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 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 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 is a prime number, then for any integer a, then $a^p\; \backslash equiv\; a\; \backslash pmod\; p$" is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and $\backslash phi(n)$ is Euler's totient function, then $a^\{\backslash varphi\; (n)\}\; \backslash equiv\; 1\; \backslash pmod\{n\}$", in the case that is a prime number.
• Euler's identity $e^\{i\; \backslash pi\}\; =\; -1$ is a special case of Euler's formula which states "for any real number x: $e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x$", in the case that = $\backslash pi$.