Special case

Template:Short description In logic, especially as applied in mathematics, concept Template:Mvar is a special case or specialization of concept Template:Mvar precisely if every instance of Template:Mvar is also an instance of Template:Mvar but not vice versa, or equivalently, if Template:Mvar is a generalization of Template:Mvar. 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 Template:Mvar is true, one can immediately deduce that Template:Mvar is true as well, and if Template:Mvar is false, Template:Mvar 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.
• Fermat's Last Theorem, that Template:Mvar has no solutions in positive integers with Template:Mvar, is a special case of Beal's conjecture, that Template:Mvar has no primitive solutions in positive integers with Template:Mvar, Template:Mvar, and Template:Mvar all greater than 2, specifically, the case of Template:Mvar.
• 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 Template:Mvar is a prime number, then for any integer a, then $a^p\equiv a(\mathrm{mod}p)$" is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and $\varphi (n)$ is Euler's totient function, then $a^{\phi (n)}\equiv 1(\mathrm{mod}n)$", in the case that Template:Mvar 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 Template:Mvar = $\pi$.